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<span>Concepts</span> </div> </a> <button aria-controls="toc-Concepts-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Concepts subsection</span> </button> <ul id="toc-Concepts-sublist" class="vector-toc-list"> <li id="toc-Variables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Variables</span> </div> </a> <ul id="toc-Variables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simplifying_expressions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simplifying_expressions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Simplifying expressions</span> </div> </a> <ul id="toc-Simplifying_expressions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Equations</span> </div> </a> <ul id="toc-Equations-sublist" class="vector-toc-list"> <li id="toc-Properties_of_equality" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Properties_of_equality"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.1</span> <span>Properties of equality</span> </div> </a> <ul id="toc-Properties_of_equality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties_of_inequality" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Properties_of_inequality"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.2</span> <span>Properties of inequality</span> </div> </a> <ul id="toc-Properties_of_inequality-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Substitution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Substitution"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Substitution</span> </div> </a> <ul id="toc-Substitution-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Solving_algebraic_equations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Solving_algebraic_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Solving algebraic equations</span> </div> </a> <button aria-controls="toc-Solving_algebraic_equations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Solving algebraic equations subsection</span> </button> <ul id="toc-Solving_algebraic_equations-sublist" class="vector-toc-list"> <li id="toc-Linear_equations_with_one_variable" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linear_equations_with_one_variable"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Linear equations with one variable</span> </div> </a> <ul id="toc-Linear_equations_with_one_variable-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linear_equations_with_two_variables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linear_equations_with_two_variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Linear equations with two variables</span> </div> </a> <ul id="toc-Linear_equations_with_two_variables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quadratic_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quadratic_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Quadratic equations</span> </div> </a> <ul id="toc-Quadratic_equations-sublist" class="vector-toc-list"> <li id="toc-Complex_numbers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Complex_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.1</span> <span>Complex numbers</span> </div> </a> <ul id="toc-Complex_numbers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Exponential_and_logarithmic_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exponential_and_logarithmic_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Exponential and logarithmic equations</span> </div> </a> <ul id="toc-Exponential_and_logarithmic_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Radical_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Radical_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Radical equations</span> </div> </a> <ul id="toc-Radical_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-System_of_linear_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#System_of_linear_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>System of linear equations</span> </div> </a> <ul id="toc-System_of_linear_equations-sublist" class="vector-toc-list"> <li id="toc-Elimination_method" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Elimination_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6.1</span> <span>Elimination method</span> </div> </a> <ul id="toc-Elimination_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Substitution_method" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Substitution_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6.2</span> <span>Substitution method</span> </div> </a> <ul id="toc-Substitution_method-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_types_of_systems_of_linear_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_types_of_systems_of_linear_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Other types of systems of linear equations</span> </div> </a> <ul id="toc-Other_types_of_systems_of_linear_equations-sublist" class="vector-toc-list"> <li id="toc-Inconsistent_systems" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Inconsistent_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7.1</span> <span>Inconsistent systems</span> </div> </a> <ul id="toc-Inconsistent_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Undetermined_systems" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Undetermined_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7.2</span> <span>Undetermined systems</span> </div> </a> <ul id="toc-Undetermined_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Over-_and_underdetermined_systems" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Over-_and_underdetermined_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7.3</span> <span>Over- and underdetermined systems</span> </div> </a> <ul id="toc-Over-_and_underdetermined_systems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Elementary algebra</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 56 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-56" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">56 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Elementare_Algebra" title="Elementare Algebra – Alemannic" lang="gsw" hreflang="gsw" data-title="Elementare Algebra" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D8%A8%D8%B1_%D8%A7%D8%A8%D8%AA%D8%AF%D8%A7%D8%A6%D9%8A" title="جبر ابتدائي – Arabic" lang="ar" hreflang="ar" data-title="جبر ابتدائي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/%C3%81lxebra_elemental" title="Álxebra elemental – Asturian" lang="ast" hreflang="ast" data-title="Álxebra elemental" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A7%8D%E0%A6%B0%E0%A6%BE%E0%A6%A5%E0%A6%AE%E0%A6%BF%E0%A6%95_%E0%A6%AC%E0%A7%80%E0%A6%9C%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4" title="প্রাথমিক বীজগণিত – Bangla" lang="bn" hreflang="bn" data-title="প্রাথমিক বীজগণিত" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%AD%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D0%B0%D1%80_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Элементар алгебра – Bashkir" lang="ba" hreflang="ba" data-title="Элементар алгебра" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%AD%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D0%B0%D1%80%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Элементарная алгебра – Belarusian" lang="be" hreflang="be" data-title="Элементарная алгебра" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%AD%D0%BB%D0%B5%D0%BC%D1%8D%D0%BD%D1%82%D0%B0%D1%80%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BB%D1%8C%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Элемэнтарная альгебра – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Элемэнтарная альгебра" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%95%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D0%B0%D1%80%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Елементарна алгебра – Bulgarian" lang="bg" hreflang="bg" data-title="Елементарна алгебра" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Elementarna_algebra" title="Elementarna algebra – Bosnian" lang="bs" hreflang="bs" data-title="Elementarna algebra" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/%C3%80lgebra_elemental" title="Àlgebra elemental – Catalan" lang="ca" hreflang="ca" data-title="Àlgebra elemental" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%AD%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D0%B0%D1%80%D0%BB%C4%83_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Элементарлă алгебра – Chuvash" lang="cv" hreflang="cv" data-title="Элементарлă алгебра" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Element%C3%A1rn%C3%AD_algebra" title="Elementární algebra – Czech" lang="cs" hreflang="cs" data-title="Elementární algebra" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Elementare_Algebra" title="Elementare Algebra – German" lang="de" hreflang="de" data-title="Elementare Algebra" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%84%CE%BF%CE%B9%CF%87%CE%B5%CE%B9%CF%8E%CE%B4%CE%B7%CF%82_%CE%AC%CE%BB%CE%B3%CE%B5%CE%B2%CF%81%CE%B1" title="Στοιχειώδης άλγεβρα – Greek" lang="el" hreflang="el" data-title="Στοιχειώδης άλγεβρα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%81lgebra_elemental" title="Álgebra elemental – Spanish" lang="es" hreflang="es" data-title="Álgebra elemental" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Baza_algebro" title="Baza algebro – Esperanto" lang="eo" hreflang="eo" data-title="Baza algebro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Oinarrizko_aljebra" title="Oinarrizko aljebra – Basque" lang="eu" hreflang="eu" data-title="Oinarrizko aljebra" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AC%D8%A8%D8%B1_%D9%85%D9%82%D8%AF%D9%85%D8%A7%D8%AA%DB%8C" title="جبر مقدماتی – Persian" lang="fa" hreflang="fa" data-title="جبر مقدماتی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Alg%C3%A8bre_classique" title="Algèbre classique – French" lang="fr" hreflang="fr" data-title="Algèbre classique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/%C3%81lxebra_elemental" title="Álxebra elemental – Galician" lang="gl" hreflang="gl" data-title="Álxebra elemental" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B4%88%EB%93%B1%EB%8C%80%EC%88%98%ED%95%99" title="초등대수학 – Korean" lang="ko" hreflang="ko" data-title="초등대수학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8F%D5%A1%D6%80%D6%80%D5%A1%D5%AF%D5%A1%D5%B6_%D5%B0%D5%A1%D5%B6%D6%80%D5%A1%D5%B0%D5%A1%D5%B7%D5%AB%D5%BE" title="Տարրական հանրահաշիվ – Armenian" lang="hy" hreflang="hy" data-title="Տարրական հանրահաշիվ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%B0%E0%A4%AE%E0%A5%8D%E0%A4%AD%E0%A4%BF%E0%A4%95_%E0%A4%AC%E0%A5%80%E0%A4%9C%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4" title="प्रारम्भिक बीजगणित – Hindi" lang="hi" hreflang="hi" data-title="प्रारम्भिक बीजगणित" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Aljabar_elementer" title="Aljabar elementer – Indonesian" lang="id" hreflang="id" data-title="Aljabar elementer" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Algebra_elementari" title="Algebra elementari – Interlingua" lang="ia" hreflang="ia" data-title="Algebra elementari" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Algebra_elementare" title="Algebra elementare – Italian" lang="it" hreflang="it" data-title="Algebra elementare" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%91%D7%A1%D7%99%D7%A1%D7%99%D7%AA" title="אלגברה בסיסית – Hebrew" lang="he" hreflang="he" data-title="אלגברה בסיסית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%9E%E0%BA%B6%E0%BA%94%E0%BA%8A%E0%BA%B0%E0%BA%84%E0%BA%B0%E0%BA%99%E0%BA%B4%E0%BA%94%E0%BA%9E%E0%BA%B7%E0%BB%89%E0%BA%99%E0%BA%96%E0%BA%B2%E0%BA%99" title="ພຶດຊະຄະນິດພື້ນຖານ – Lao" lang="lo" hreflang="lo" data-title="ພຶດຊະຄະນິດພື້ນຖານ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Algebra_elementaris" title="Algebra elementaris – Latin" lang="la" hreflang="la" data-title="Algebra elementaris" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Aljebra_fundal" title="Aljebra fundal – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Aljebra fundal" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Elemi_algebra" title="Elemi algebra – Hungarian" lang="hu" hreflang="hu" data-title="Elemi algebra" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%95%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D0%B0%D1%80%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Елементарна алгебра – Macedonian" lang="mk" hreflang="mk" data-title="Елементарна алгебра" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Algebra_permulaan" title="Algebra permulaan – Malay" lang="ms" hreflang="ms" data-title="Algebra permulaan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Aljabar_elementer" title="Aljabar elementer – Minangkabau" lang="min" hreflang="min" data-title="Aljabar elementer" data-language-autonym="Minangkabau" data-language-local-name="Minangkabau" class="interlanguage-link-target"><span>Minangkabau</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Elementaire_algebra" title="Elementaire algebra – Dutch" lang="nl" hreflang="nl" data-title="Elementaire algebra" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%88%9D%E7%AD%89%E4%BB%A3%E6%95%B0%E5%AD%A6" title="初等代数学 – Japanese" lang="ja" hreflang="ja" data-title="初等代数学" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Element%C3%A6r_algebra" title="Elementær algebra – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Elementær algebra" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Alg%C3%A8bra_element%C3%A0ria" title="Algèbra elementària – Occitan" lang="oc" hreflang="oc" data-title="Algèbra elementària" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%A7%D8%A8%D8%AA%D8%AF%D8%A7%D8%A6%DB%8C_%D8%A7%D9%84%D8%AC%D8%A8%D8%B1%D8%A7" title="ابتدائی الجبرا – Western Punjabi" lang="pnb" hreflang="pnb" data-title="ابتدائی الجبرا" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%81lgebra_elementar" title="Álgebra elementar – Portuguese" lang="pt" hreflang="pt" data-title="Álgebra elementar" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%AD%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D0%B0%D1%80%D0%BD%D0%B0%D1%8F_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Элементарная алгебра – Russian" lang="ru" hreflang="ru" data-title="Элементарная алгебра" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B8%E0%B7%96%E0%B6%BD%E0%B7%92%E0%B6%9A_%E0%B7%80%E0%B7%93%E0%B6%A2_%E0%B6%9C%E0%B6%AB%E0%B7%92%E0%B6%AD%E0%B6%BA" title="මූලික වීජ ගණිතය – Sinhala" lang="si" hreflang="si" data-title="මූලික වීජ ගණිතය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Elementary_algebra" title="Elementary algebra – Simple English" lang="en-simple" hreflang="en-simple" data-title="Elementary algebra" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AC%DB%95%D8%A8%D8%B1%DB%8C_%D8%B3%DB%95%D8%B1%DB%95%D8%AA%D8%A7%DB%8C%DB%8C" title="جەبری سەرەتایی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="جەبری سەرەتایی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%95%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D0%B0%D1%80%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Елементарна алгебра – Serbian" lang="sr" hreflang="sr" data-title="Елементарна алгебра" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Element%C3%A4r_algebra" title="Elementär algebra – Swedish" lang="sv" hreflang="sv" data-title="Elementär algebra" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Alhebrang_elementaryo" title="Alhebrang elementaryo – Tagalog" lang="tl" hreflang="tl" data-title="Alhebrang elementaryo" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%85%E0%AE%9F%E0%AE%BF%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%9F%E0%AF%88_%E0%AE%87%E0%AE%AF%E0%AE%B1%E0%AF%8D%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D" title="அடிப்படை இயற்கணிதம் – Tamil" lang="ta" hreflang="ta" data-title="அடிப்படை இயற்கணிதம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Temel_cebir" title="Temel cebir – Turkish" lang="tr" hreflang="tr" data-title="Temel cebir" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%95%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D0%B0%D1%80%D0%BD%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0" title="Елементарна алгебра – Ukrainian" lang="uk" hreflang="uk" data-title="Елементарна алгебра" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%A7%D8%A8%D8%AA%D8%AF%D8%A7%D8%A6%DB%8C_%D8%A7%D9%84%D8%AC%D8%A8%D8%B1%D8%A7" title="ابتدائی الجبرا – Urdu" lang="ur" hreflang="ur" data-title="ابتدائی الجبرا" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BA%A1i_s%E1%BB%91_s%C6%A1_c%E1%BA%A5p" title="Đại số sơ cấp – Vietnamese" lang="vi" hreflang="vi" data-title="Đại số sơ cấp" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%88%9D%E7%AD%89%E4%BB%A3%E6%95%B0" title="初等代数 – Wu" lang="wuu" hreflang="wuu" data-title="初等代数" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A2%D7%9C%D7%A2%D7%9E%D7%A2%D7%A0%D7%98%D7%90%D7%A8%D7%A2_%D7%90%D7%9C%D7%92%D7%A2%D7%91%D7%A8%D7%A2" title="עלעמענטארע אלגעברע – Yiddish" lang="yi" hreflang="yi" data-title="עלעמענטארע אלגעברע" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%9F%BA%E6%9C%AC%E4%BB%A3%E6%95%B8" title="基本代數 – Cantonese" lang="yue" hreflang="yue" data-title="基本代數" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%88%9D%E7%AD%89%E4%BB%A3%E6%95%B8" title="初等代數 – Chinese" lang="zh" hreflang="zh" data-title="初等代數" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q211294#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav 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class="thumb tright" style=""><div class="thumbinner" style="width:202px"><div class="thumbimage noresize" style="width:200px;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overset {}{\underset {}{x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <munder> <mrow> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>b</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mi>c</mi> </msqrt> </mrow> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> </mrow> <mrow /> </munder> <mrow /> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overset {}{\underset {}{x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/575def143df0130b657f403d90eefe4484e1e687" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:21.525ex; height:8.676ex;" alt="{\displaystyle {\overset {}{\underset {}{x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}}}}"></span></div><div class="thumbcaption">The <a href="/wiki/Quadratic_formula" title="Quadratic formula">quadratic formula</a>, which is the solution to the <a href="/wiki/Quadratic_equation" title="Quadratic equation">quadratic equation</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax^{2}+bx+c=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax^{2}+bx+c=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23e70cfa003f402d108ec04d97983fb62f69536e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.89ex; height:2.843ex;" alt="{\displaystyle ax^{2}+bx+c=0}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f455a7f96d74aa94573d8e32da3b240ab0aa294f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.491ex; height:2.676ex;" alt="{\displaystyle a\neq 0}"></span>. Here the symbols <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>, and <span class="texhtml mvar" style="font-style:italic;">c</span> represent arbitrary numbers, and <span class="texhtml mvar" style="font-style:italic;">x</span> is a variable which represents the solution of the equation.</div></div></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Polynomialdeg2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Polynomialdeg2.svg/200px-Polynomialdeg2.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Polynomialdeg2.svg/300px-Polynomialdeg2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Polynomialdeg2.svg/400px-Polynomialdeg2.svg.png 2x" data-file-width="320" data-file-height="320" /></a><figcaption>Two-dimensional plot (red curve) of the algebraic equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x^{2}-x-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x^{2}-x-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61efa8d731eb6affd257bb15138b7bef53e5ed05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.811ex; height:3.009ex;" alt="{\displaystyle y=x^{2}-x-2}"></span>.</figcaption></figure> <p><b>Elementary algebra</b>, also known as <b>college algebra</b>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> encompasses the basic concepts of <a href="/wiki/Algebra" title="Algebra">algebra</a>. It is often contrasted with <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a>: arithmetic deals with specified <a href="/wiki/Number" title="Number">numbers</a>,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> whilst algebra introduces <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a> (quantities without fixed values).<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>This use of variables entails use of algebraic notation and an understanding of the general rules of the <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a> introduced in arithmetic: addition, subtraction, multiplication, division, etc. Unlike <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>, elementary algebra is not concerned with <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structures</a> outside the realm of <a href="/wiki/Real_number" title="Real number">real</a> and <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>. </p><p>It is typically taught to <a href="/wiki/Secondary_school" title="Secondary school">secondary school</a> students and at introductory college level in the <a href="/wiki/United_States" title="United States">United States</a>,<sup id="cite_ref-leff_4-0" class="reference"><a href="#cite_note-leff-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and builds on their understanding of <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a>. The use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of problems. Many quantitative relationships in <a href="/wiki/Science" title="Science">science</a> and mathematics are expressed as algebraic <a href="/wiki/Equation" title="Equation">equations</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Algebraic_operations">Algebraic operations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=1" title="Edit section: Algebraic operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Algebraic_operation" title="Algebraic operation">Algebraic operation</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Algebraic_operation&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Quadratic_root.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Quadratic_root.svg/220px-Quadratic_root.svg.png" decoding="async" width="220" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Quadratic_root.svg/330px-Quadratic_root.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Quadratic_root.svg/440px-Quadratic_root.svg.png 2x" data-file-width="1423" data-file-height="450" /></a><figcaption>Algebraic operations in the solution to the <a href="/wiki/Quadratic_equation" title="Quadratic equation">quadratic equation</a>. The radical sign √, denoting a <a href="/wiki/Square_root" title="Square root">square root</a>, is equivalent to <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a> to the power of <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>. The <a href="/wiki/Plus%E2%80%93minus_sign" title="Plus–minus sign">± sign</a> means the <a href="/wiki/Equation" title="Equation">equation</a> can be written with either a + or a – sign.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a basic <a href="/wiki/Algebraic_operation" title="Algebraic operation">algebraic operation</a> is any one of the common <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a> of elementary algebra, which include <a href="/wiki/Addition" title="Addition">addition</a>, <a href="/wiki/Subtraction" title="Subtraction">subtraction</a>, <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a>, raising to a whole number <a href="/wiki/Exponentiation" title="Exponentiation">power</a>, and taking <a href="/wiki/Nth_root" title="Nth root">roots</a> (<a href="/wiki/Fraction" title="Fraction">fractional</a> power).<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> These operations may be performed on <a href="/wiki/Number" title="Number">numbers</a>, in which case they are often called <i><a href="/wiki/Arithmetic_operations" class="mw-redirect" title="Arithmetic operations">arithmetic operations</a></i>. They may also be performed, in a similar way, on <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a>, <a href="/wiki/Algebraic_expression" title="Algebraic expression">algebraic expressions</a>,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> and more generally, on elements of <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structures</a>, such as <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a> and <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> An algebraic operation may also be defined more generally as a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> from a <a href="/wiki/Cartesian_product#n-ary_Cartesian_power" title="Cartesian product">Cartesian power</a> from a given <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> to the same set.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> The term <i>algebraic operation</i> may also be used for operations that may be defined by compounding basic algebraic operations, such as the <a href="/wiki/Dot_product" title="Dot product">dot product</a>. In <a href="/wiki/Calculus" title="Calculus">calculus</a> and <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, <i>algebraic operation</i> is also used for the operations that may be defined by purely <a href="/wiki/Algebra" title="Algebra">algebraic methods</a>. For example, <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a> with an <a href="/wiki/Integer" title="Integer">integer</a> or <a href="/wiki/Rational_number" title="Rational number">rational</a> exponent is an algebraic operation, but not the general exponentiation with a <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex</a> exponent. Also, the <a href="/wiki/Derivative" title="Derivative">derivative</a> is an operation that is not algebraic.</div></div> <div class="mw-heading mw-heading2"><h2 id="Algebraic_notation">Algebraic notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=2" title="Edit section: Algebraic notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Mathematical_notation" title="Mathematical notation">Mathematical notation</a></div> <p>Algebraic notation describes the rules and conventions for writing <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">mathematical expressions</a>, as well as the terminology used for talking about parts of expressions. For example, the expression <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3x^{2}-2xy+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x^{2}-2xy+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e64f206e30ab6fe182a8aa38cbd30353bbfbb202" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.671ex; width:13.882ex; height:3.009ex;" aria-hidden="true" alt="{\displaystyle 3x^{2}-2xy+c}"></span> has the following components: </p> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Algebraic_equation_notation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Algebraic_equation_notation.svg/256px-Algebraic_equation_notation.svg.png" decoding="async" width="256" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Algebraic_equation_notation.svg/384px-Algebraic_equation_notation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/be/Algebraic_equation_notation.svg/512px-Algebraic_equation_notation.svg.png 2x" data-file-width="883" data-file-height="516" /></a><figcaption><div><ol><li><a href="/wiki/Exponent" class="mw-redirect" title="Exponent">exponent</a> (power)</li><li><a href="/wiki/Coefficient" title="Coefficient">coefficient</a></li><li><a href="/wiki/Addend" class="mw-redirect" title="Addend">term</a></li><li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operation</a></li><li><a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constant</a>,<br /><span class="texhtml mvar" style="font-style:italic;">x, y</span>. <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a></li></ol></div></figcaption></figure> <p>A <i>coefficient</i> is a numerical value, or letter representing a numerical constant, that multiplies a variable (the operator is omitted). A <i>term</i> is an <a href="/wiki/Addend#Notation_and_terminology" class="mw-redirect" title="Addend">addend or a summand</a>, a group of coefficients, variables, constants and exponents that may be separated from the other terms by the plus and minus operators.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Letters represent variables and constants. By convention, letters at the beginning of the alphabet (e.g. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}"></span>) are typically used to represent <a href="/wiki/Mathematical_constant" title="Mathematical constant">constants</a>, and those toward the end of the alphabet (e.g. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea0abffd33a692ded22accc104515a032851dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.009ex;" alt="{\displaystyle x,y}"></span> and <span class="texhtml mvar" style="font-style:italic;">z</span>) are used to represent <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> They are usually printed in italics.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Algebraic_operation" title="Algebraic operation">Algebraic operations</a> work in the same way as <a href="/wiki/Arithmetic_operations" class="mw-redirect" title="Arithmetic operations">arithmetic operations</a>,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> such as <a href="/wiki/Addition" title="Addition">addition</a>, <a href="/wiki/Subtraction" title="Subtraction">subtraction</a>, <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a> and <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a>,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> and are applied to algebraic variables and terms. Multiplication symbols are usually omitted, and implied when there is no space between two variables or terms, or when a <a href="/wiki/Coefficient" title="Coefficient">coefficient</a> is used. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\times x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>×</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\times x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeb4d4da2ddd72326baad3dd527059a7350fb770" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.338ex; width:6.387ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle 3\times x^{2}}"></span> is written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f94e37e2833e124c2339e013806d641bba54f4ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.338ex; width:3.546ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle 3x^{2}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times x\times y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mi>x</mi> <mo>×</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times x\times y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e83c3f914bae95e8d54fe8ab8b1c9886ff287dea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.328ex; height:2.509ex;" alt="{\displaystyle 2\times x\times y}"></span> may be written <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2xy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf28a82088fc2021c794fee1de241f90622d5ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.648ex; height:2.509ex;" alt="{\displaystyle 2xy}"></span>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>Usually terms with the highest power (<a href="/wiki/Exponentiation" title="Exponentiation">exponent</a>), are written on the left, for example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle x^{2}}"></span> is written to the left of <span class="texhtml mvar" style="font-style:italic;">x</span>. When a coefficient is one, it is usually omitted (e.g. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11216e2eac7b4b023de3753b8ba582dd7324014a" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.338ex; width:3.546ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle 1x^{2}}"></span> is written <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle x^{2}}"></span>).<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Likewise when the exponent (power) is one, (e.g. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3x^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/823cf71efe5336a27a8dc9e40bbea234098d8734" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.338ex; width:3.546ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle 3x^{1}}"></span> is written <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd854d1deca171ed607f4a2f326c3a0ee3029f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.338ex; width:2.492ex; height:2.176ex;" aria-hidden="true" alt="{\displaystyle 3x}"></span>).<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> When the exponent is zero, the result is always 1 (e.g. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1871ffeb57c11624b375dbb7157d5887c706eb87" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle x^{0}}"></span> is always rewritten to <span class="texhtml mvar" style="font-style:italic;">1</span>).<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> However <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/106f0c4e1cbccbfcbb61001a8c17b8427c65366d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 0^{0}}"></span>, being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents. </p> <div class="mw-heading mw-heading3"><h3 id="Alternative_notation">Alternative notation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=3" title="Edit section: Alternative notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Other types of notation are used in algebraic expressions when the required formatting is not available, or can not be implied, such as where only letters and symbols are available. As an illustration of this, while exponents are usually formatted using superscripts, e.g., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle x^{2}}"></span>, in <a href="/wiki/Plain_text" title="Plain text">plain text</a>, and in the <a href="/wiki/TeX" title="TeX">TeX</a> mark-up language, the <a href="/wiki/Caret" title="Caret">caret</a> symbol <span class="nounderlines" style="border: 1px solid var(--border-color-muted,#ddd); color: var(--color-base); background-color: var( --background-color-neutral-subtle, #fdfdfd); padding: 1px 1px;">^</span> represents exponentiation, so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle x^{2}}"></span> is written as "x^2".<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> This also applies to some programming languages such as Lua. In programming languages such as <a href="/wiki/Ada_(programming_language)" title="Ada (programming language)">Ada</a>,<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Fortran" title="Fortran">Fortran</a>,<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Perl" title="Perl">Perl</a>,<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Ruby_(programming_language)" title="Ruby (programming language)">Ruby</a>,<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> a double asterisk is used, so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle x^{2}}"></span> is written as "x**2". Many programming languages and calculators use a single asterisk to represent the multiplication symbol,<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> and it must be explicitly used, for example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd854d1deca171ed607f4a2f326c3a0ee3029f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" style="vertical-align: -0.338ex; width:2.492ex; height:2.176ex;" aria-hidden="true" alt="{\displaystyle 3x}"></span> is written "3*x". </p> <div class="mw-heading mw-heading2"><h2 id="Concepts">Concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=4" title="Edit section: Concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Variables">Variables</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=5" title="Edit section: Variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Pi-equals-circumference-over-diametre.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Pi-equals-circumference-over-diametre.svg/220px-Pi-equals-circumference-over-diametre.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Pi-equals-circumference-over-diametre.svg/330px-Pi-equals-circumference-over-diametre.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/Pi-equals-circumference-over-diametre.svg/440px-Pi-equals-circumference-over-diametre.svg.png 2x" data-file-width="590" data-file-height="590" /></a><figcaption>Example of variables showing the relationship between a circle's diameter and its circumference. For any <a href="/wiki/Circle" title="Circle">circle</a>, its <a href="/wiki/Circumference" title="Circumference">circumference</a> <span class="texhtml mvar" style="font-style:italic;">c</span>, divided by its <a href="/wiki/Diameter" title="Diameter">diameter</a> <span class="texhtml mvar" style="font-style:italic;">d</span>, is equal to the constant <a href="/wiki/Pi" title="Pi">pi</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> (approximately 3.14).</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">Variable (mathematics)</a></div> <p>Elementary algebra builds on and extends arithmetic<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> by introducing letters called variables to represent general (non-specified) numbers. This is useful for several reasons. </p> <ol><li><b>Variables may represent numbers whose values are not yet known</b>. For example, if the temperature of the current day, C, is 20 degrees higher than the temperature of the previous day, P, then the problem can be described algebraically as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=P+20}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mi>P</mi> <mo>+</mo> <mn>20</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=P+20}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5687cadd1fdb6197673e771331876c11b1013ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.776ex; height:2.343ex;" alt="{\displaystyle C=P+20}"></span>.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup></li> <li><b>Variables allow one to describe <i>general</i> problems,<sup id="cite_ref-leff_4-1" class="reference"><a href="#cite_note-leff-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> without specifying the values of the quantities that are involved.</b> For example, it can be stated specifically that 5 minutes is equivalent to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 60\times 5=300}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>60</mn> <mo>×</mo> <mn>5</mn> <mo>=</mo> <mn>300</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 60\times 5=300}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/939c7ab17c10efa4dfaed57bcddc1b6e24660a74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.914ex; height:2.176ex;" alt="{\displaystyle 60\times 5=300}"></span> seconds. A more general (algebraic) description may state that the number of seconds, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=60\times m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mn>60</mn> <mo>×</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=60\times m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6634f8738ffb1a017147fa9ffbef3134dcb1f68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.395ex; height:2.176ex;" alt="{\displaystyle s=60\times m}"></span>, where m is the number of minutes.</li> <li><b>Variables allow one to describe mathematical relationships between quantities that may vary.</b><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> For example, the relationship between the circumference, <i>c</i>, and diameter, <i>d</i>, of a circle is described by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =c/d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =c/d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd2d6b434acea51ab783f150892bc3d83c7269a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.816ex; height:2.843ex;" alt="{\displaystyle \pi =c/d}"></span>.</li> <li><b>Variables allow one to describe some mathematical properties.</b> For example, a basic property of addition is <a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">commutativity</a> which states that the order of numbers being added together does not matter. Commutativity is stated algebraically as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+b)=(b+a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b)=(b+a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceb57a40ce722ec9b9f22fb3d7dd2647342c87cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.853ex; height:2.843ex;" alt="{\displaystyle (a+b)=(b+a)}"></span>.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup></li></ol> <div class="mw-heading mw-heading3"><h3 id="Simplifying_expressions">Simplifying expressions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=6" title="Edit section: Simplifying expressions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression (mathematics)</a> and <a href="/wiki/Computer_algebra#Simplification" title="Computer algebra">Computer algebra § Simplification</a></div> <p>Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations (<a href="/wiki/Addition" title="Addition">addition</a>, <a href="/wiki/Subtraction" title="Subtraction">subtraction</a>, <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a> and <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a>). For example, </p> <ul><li>Added terms are simplified using coefficients. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+x+x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+x+x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a7dd4ae2016558a94a840db44159f35300f9660" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.67ex; height:2.176ex;" alt="{\displaystyle x+x+x}"></span> can be simplified as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd854d1deca171ed607f4a2f326c3a0ee3029f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.492ex; height:2.176ex;" alt="{\displaystyle 3x}"></span> (where 3 is a numerical coefficient).</li> <li>Multiplied terms are simplified using exponents. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\times x\times x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>×</mo> <mi>x</mi> <mo>×</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\times x\times x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e4e08469ba2d19e061ed90b4d30f4b17a7cf572" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.67ex; height:1.676ex;" alt="{\displaystyle x\times x\times x}"></span> is represented as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2d4389a3b6f20cb8a118506601a68c2263143a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{3}}"></span></li> <li>Like terms are added together,<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> for example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x^{2}+3ab-x^{2}+ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mi>a</mi> <mi>b</mi> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x^{2}+3ab-x^{2}+ab}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5328816e9887a6f0081c3a0483e6869494cb17f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.069ex; height:2.843ex;" alt="{\displaystyle 2x^{2}+3ab-x^{2}+ab}"></span> is written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+4ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+4ab}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61e65e40a5d5471dae0c0d5a7588d614b904cf01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.614ex; height:2.843ex;" alt="{\displaystyle x^{2}+4ab}"></span>, because the terms containing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}"></span> are added together, and, the terms containing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49337c5cf256196e2292f7047cb5da68c24ca95d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.227ex; height:2.176ex;" alt="{\displaystyle ab}"></span> are added together.</li> <li>Brackets can be "multiplied out", using <a href="/wiki/Distributive_property" title="Distributive property">the distributive property</a>. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(2x+3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(2x+3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80fffddd92727a5c5d71c7583af8ca32afb30d4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.634ex; height:2.843ex;" alt="{\displaystyle x(2x+3)}"></span> can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x\times 2x)+(x\times 3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>×</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>×</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x\times 2x)+(x\times 3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1e63c90413397ab755981d0f2e37c34f05f9076" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.454ex; height:2.843ex;" alt="{\displaystyle (x\times 2x)+(x\times 3)}"></span> which can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x^{2}+3x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x^{2}+3x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51b504e73b6e1812fe65d1f827d0bd08c3fcfe5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.879ex; height:2.843ex;" alt="{\displaystyle 2x^{2}+3x}"></span></li> <li>Expressions can be <a href="/wiki/Factorization" title="Factorization">factored</a>. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6x^{5}+3x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6x^{5}+3x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81b597d7c2a7ccb6943812c240ffd627217a4e6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.933ex; height:2.843ex;" alt="{\displaystyle 6x^{5}+3x^{2}}"></span>, by dividing both terms by the common <a href="/wiki/Factor_(arithmetic)" class="mw-redirect" title="Factor (arithmetic)">factor</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f94e37e2833e124c2339e013806d641bba54f4ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.546ex; height:2.676ex;" alt="{\displaystyle 3x^{2}}"></span> can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3x^{2}(2x^{3}+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x^{2}(2x^{3}+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d7db4c92ee8fcdd3b2e806fb9372f7b500d5c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.905ex; height:3.176ex;" alt="{\displaystyle 3x^{2}(2x^{3}+1)}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Equations">Equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=7" title="Edit section: Equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Pythagorean_theorem_-_Ani.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Pythagorean_theorem_-_Ani.gif/220px-Pythagorean_theorem_-_Ani.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Pythagorean_theorem_-_Ani.gif/330px-Pythagorean_theorem_-_Ani.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Pythagorean_theorem_-_Ani.gif/440px-Pythagorean_theorem_-_Ani.gif 2x" data-file-width="600" data-file-height="600" /></a><figcaption>Animation illustrating <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagoras' rule</a> for a right-angle triangle, which shows the algebraic relationship between the triangle's hypotenuse, and the other two sides.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Equation" title="Equation">Equation</a></div> <p>An equation states that two expressions are equal using the symbol for equality, = (the <a href="/wiki/Equals_sign" title="Equals sign">equals sign</a>).<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> One of the best-known equations describes Pythagoras' law relating the length of the sides of a <a href="/wiki/Right_angle" title="Right angle">right angle</a> triangle:<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{2}=a^{2}+b^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}=a^{2}+b^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bace667e8cfea3ef573af86a1f3e72984b10755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.336ex; height:2.843ex;" alt="{\displaystyle c^{2}=a^{2}+b^{2}}"></span></dd></dl> <p>This equation states that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f3386a00382ce857fb0b3b04b9fa2bbe5cfae9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.061ex; height:2.676ex;" alt="{\displaystyle c^{2}}"></span>, representing the square of the length of the side that is the hypotenuse, the side opposite the right angle, is equal to the sum (addition) of the squares of the other two sides whose lengths are represented by <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span>. </p><p>An equation is the claim that two expressions have the same value and are equal. Some equations are true for all values of the involved variables (such as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b=b+a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b=b+a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/684f43b5094501674e8314be5e24a80ee64682e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.234ex; height:2.343ex;" alt="{\displaystyle a+b=b+a}"></span>); such equations are called <a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">identities</a>. Conditional equations are true for only some values of the involved variables, e.g. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}-1=8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}-1=8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33d5fd32633b72ead0e1597272822a628a0cfa84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.648ex; height:2.843ex;" alt="{\displaystyle x^{2}-1=8}"></span> is true only for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/871a5063af170fa536b144fbcc5745146a42cc13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=3}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=-3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=-3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f2b353db7dfc28d1d80b789ffc2e4a63c3a2a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.399ex; height:2.343ex;" alt="{\displaystyle x=-3}"></span>. The values of the variables which make the equation true are the solutions of the equation and can be found through <a href="/wiki/Equation_solving" title="Equation solving">equation solving</a>. </p><p>Another type of equation is inequality. Inequalities are used to show that one side of the equation is greater, or less, than the other. The symbols used for this are: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83fc0063781fb9bf4ec7608b2fd11ed6d5b05a13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a>b}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle >}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle >}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b27b77ab4e3293ea9ce65cef60fea655c398423" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle >}"></span> represents 'greater than', and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle <}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo><</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle <}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33737c89a17785dacc8638b4d66db3d5c8670de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle <}"></span> represents 'less than'. Just like standard equality equations, numbers can be added, subtracted, multiplied or divided. The only exception is that when multiplying or dividing by a negative number, the inequality symbol must be flipped. </p> <div class="mw-heading mw-heading4"><h4 id="Properties_of_equality">Properties of equality</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=8" title="Edit section: Properties of equality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By definition, equality is an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a>, meaning it is <a href="/wiki/Reflexive_relation" title="Reflexive relation">reflexive</a> (i.e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e808eb0d800bf37c7707d0ded5d243f58459a1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.094ex; height:2.176ex;" alt="{\displaystyle b=b}"></span>), <a href="/wiki/Symmetric_relation" title="Symmetric relation">symmetric</a> (i.e. if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a=b}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b2e4888d97a754d4bfa4da297b226788a73c6b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle b=a}"></span>), and <a href="/wiki/Transitive_relation" title="Transitive relation">transitive</a> (i.e. if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a=b}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b133a00dc90e54130a96482c99750f845feb955e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.103ex; height:2.176ex;" alt="{\displaystyle b=c}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1beb3f1b1ad87e99791ba713839204a88b27239a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.335ex; height:1.676ex;" alt="{\displaystyle a=c}"></span>).<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> It also satisfies the important property that if two symbols are used for equal things, then one symbol can be substituted for the other in any true statement about the first and the statement will remain true. This implies the following properties: </p> <ul><li>if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a=b}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8498d2878942ddeef7d6a8ec870959f0d38d32e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.321ex; height:2.176ex;" alt="{\displaystyle c=d}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+c=b+d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+c=b+d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81ef6bd66c184b2f27a91e0bcac1d41fb2073063" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.229ex; height:2.343ex;" alt="{\displaystyle a+c=b+d}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ac=bd}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>c</mi> <mo>=</mo> <mi>b</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ac=bd}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7585cd9fe78dd1fb4e21d6528fb92f260664e9cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.549ex; height:2.176ex;" alt="{\displaystyle ac=bd}"></span>;</li> <li>if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a=b}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+c=b+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+c=b+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a8d7c402d7bd2bba2f1538877a9f92b458c4f68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.02ex; height:2.343ex;" alt="{\displaystyle a+c=b+c}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ac=bc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>c</mi> <mo>=</mo> <mi>b</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ac=bc}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d280528b8755df793b774265a3be5c4569814fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.34ex; height:2.176ex;" alt="{\displaystyle ac=bc}"></span>;</li> <li>more generally, for any function <span class="texhtml mvar" style="font-style:italic;">f</span>, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a=b}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a)=f(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(a)=f(b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70209bf143b8417feef2aed98b2e86bc8f447e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.502ex; height:2.843ex;" alt="{\displaystyle f(a)=f(b)}"></span>.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Properties_of_inequality">Properties of inequality</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=9" title="Edit section: Properties of inequality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The relations <i>less than</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle <}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo><</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle <}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33737c89a17785dacc8638b4d66db3d5c8670de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle <}"></span> and greater than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle >}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle >}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b27b77ab4e3293ea9ce65cef60fea655c398423" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle >}"></span> have the property of transitivity:<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <ul><li>If   <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}"></span>   and   <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b<c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo><</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b<c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3667e3266421420148b93988e6f7db6b8aada82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.103ex; height:2.176ex;" alt="{\displaystyle b<c}"></span>   then   <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f61bdc0f32d5cbd3c62c118ef526397ab4c1e1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.335ex; height:1.843ex;" alt="{\displaystyle a<c}"></span>;</li> <li>If   <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}"></span>   and   <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c<d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo><</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c<d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/826f129a304df73d8c5c71cc0ea8787fad9d40ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.321ex; height:2.176ex;" alt="{\displaystyle c<d}"></span>   then   <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+c<b+d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo><</mo> <mi>b</mi> <mo>+</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+c<b+d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffff35f8aa2508180de411cb546968bc63bd6c27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.229ex; height:2.343ex;" alt="{\displaystyle a+c<b+d}"></span>;<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup></li> <li>If   <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}"></span>   and   <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba126f626d61752f62eaacaf11761a54de4dc84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c>0}"></span>   then   <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ac<bc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>c</mi> <mo><</mo> <mi>b</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ac<bc}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0265433404d1a45f8a1cdda0343945d9dbed5f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.34ex; height:2.176ex;" alt="{\displaystyle ac<bc}"></span>;</li> <li>If   <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}"></span>   and   <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26a48dc798956117afd8c429c39886678c0e7204" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c<0}"></span>   then   <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle bc<ac}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mi>c</mi> <mo><</mo> <mi>a</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle bc<ac}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92b4137cd9315a7d23ed92f33ea900d97aeb95c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.34ex; height:2.176ex;" alt="{\displaystyle bc<ac}"></span>.</li></ul> <p>By reversing the inequation, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle <}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo><</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle <}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33737c89a17785dacc8638b4d66db3d5c8670de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle <}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle >}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle >}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b27b77ab4e3293ea9ce65cef60fea655c398423" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle >}"></span> can be swapped,<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> for example: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}"></span> is equivalent to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b>a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>></mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b>a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b1cfc86ca957eea4f09d683db2412a173f6f404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle b>a}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Substitution">Substitution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=10" title="Edit section: Substitution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Substitution_(algebra)" class="mw-redirect" title="Substitution (algebra)">Substitution (algebra)</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution (logic)</a></div> <p>Substitution is replacing the terms in an expression to create a new expression. Substituting 3 for <span class="texhtml mvar" style="font-style:italic;">a</span> in the expression <span class="texhtml"><i>a</i>*5</span> makes a new expression <span class="texhtml">3*5</span> with meaning <span class="texhtml">15</span>. Substituting the terms of a statement makes a new statement. When the original statement is true independently of the values of the terms, the statement created by substitutions is also true. Hence, definitions can be made in symbolic terms and interpreted through substitution: if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}:=a\times a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>:=</mo> <mi>a</mi> <mo>×</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}:=a\times a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf945c2f9fba9fdcaac892d2da199016e77112e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.329ex; height:2.676ex;" alt="{\displaystyle a^{2}:=a\times a}"></span> is meant as the definition of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de86823f6a2c1491493f6c76e8cc8560469b165c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.931ex; height:3.009ex;" alt="{\displaystyle a^{2},}"></span> as the product of <span class="texhtml mvar" style="font-style:italic;">a</span> with itself, substituting <span class="texhtml">3</span> for <span class="texhtml mvar" style="font-style:italic;">a</span> informs the reader of this statement that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84cba38173d6b69364f2016245721c333282e0d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 3^{2}}"></span> means <span class="texhtml">3 × 3 = 9</span>. Often it's not known whether the statement is true independently of the values of the terms. And, substitution allows one to derive restrictions on the possible values, or show what conditions the statement holds under. For example, taking the statement <span class="texhtml"><i>x</i> + 1 = 0</span>, if <span class="texhtml mvar" style="font-style:italic;">x</span> is substituted with <span class="texhtml">1</span>, this implies <span class="texhtml">1 + 1 = 2 = 0</span>, which is false, which implies that if <span class="texhtml"><i>x</i> + 1 = 0</span> then <span class="texhtml mvar" style="font-style:italic;">x</span> cannot be <span class="texhtml">1</span>. </p><p>If <span class="texhtml"><i>x</i></span> and <span class="texhtml"><i>y</i></span> are <a href="/wiki/Integers" class="mw-redirect" title="Integers">integers</a>, <a href="/wiki/Rationals" class="mw-redirect" title="Rationals">rationals</a>, or <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>, then <span class="texhtml"><i>xy</i> = 0</span> implies <span class="texhtml"><i>x</i> = 0</span> or <span class="texhtml"><i>y</i> = 0</span>. Consider <span class="texhtml"><i>abc</i> = 0</span>. Then, substituting <span class="texhtml"><i>a</i></span> for <span class="texhtml"><i>x</i></span> and <span class="texhtml"><i>bc</i></span> for <span class="texhtml"><i>y</i></span>, we learn <span class="texhtml"><i>a</i> = 0</span> or <span class="texhtml"><i>bc</i> = 0</span>. Then we can substitute again, letting <span class="texhtml"><i>x</i> = <i>b</i></span> and <span class="texhtml"><i>y</i> = <i>c</i></span>, to show that if <span class="texhtml"><i>bc</i> = 0</span> then <span class="texhtml"><i>b</i> = 0</span> or <span class="texhtml"><i>c</i> = 0</span>. Therefore, if <span class="texhtml"><i>abc</i> = 0</span>, then <span class="texhtml"><i>a</i> = 0</span> or (<span class="texhtml"><i>b</i> = 0</span> or <span class="texhtml"><i>c</i> = 0</span>), so <span class="texhtml"><i>abc</i> = 0</span> implies <span class="texhtml"><i>a</i> = 0</span> or <span class="texhtml"><i>b</i> = 0</span> or <span class="texhtml"><i>c</i> = 0</span>. </p><p>If the original fact were stated as "<span class="texhtml"><i>ab</i> = 0</span> implies <span class="texhtml"><i>a</i> = 0</span> or <span class="texhtml"><i>b</i> = 0</span>", then when saying "consider <span class="texhtml"><i>abc</i> = 0</span>," we would have a conflict of terms when substituting. Yet the above logic is still valid to show that if <span class="texhtml"><i>abc</i> = 0</span> then <span class="texhtml"><i>a</i> = 0</span> or <span class="texhtml"><i>b</i> = 0</span> or <span class="texhtml"><i>c</i> = 0</span> if, instead of letting <span class="texhtml"><i>a</i> = <i>a</i></span> and <span class="texhtml"><i>b</i> = <i>bc</i></span>, one substitutes <span class="texhtml"><i>a</i></span> for <span class="texhtml"><i>a</i></span> and <span class="texhtml"><i>b</i></span> for <span class="texhtml"><i>bc</i></span> (and with <span class="texhtml"><i>bc</i> = 0</span>, substituting <span class="texhtml"><i>b</i></span> for <span class="texhtml"><i>a</i></span> and <span class="texhtml"><i>c</i></span> for <span class="texhtml"><i>b</i></span>). This shows that substituting for the terms in a statement isn't always the same as letting the terms from the statement equal the substituted terms. In this situation it's clear that if we substitute an expression <span class="texhtml"><i>a</i></span> into the <span class="texhtml"><i>a</i></span> term of the original equation, the <span class="texhtml"><i>a</i></span> substituted does not refer to the <span class="texhtml"><i>a</i></span> in the statement "<span class="texhtml"><i>ab</i> = 0</span> implies <span class="texhtml"><i>a</i> = 0</span> or <span class="texhtml"><i>b</i> = 0</span>." </p> <div class="mw-heading mw-heading2"><h2 id="Solving_algebraic_equations">Solving algebraic equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=11" title="Edit section: Solving algebraic equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Equation_solving" title="Equation solving">Equation solving</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Algebraproblem.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Algebraproblem.jpg/220px-Algebraproblem.jpg" decoding="async" width="220" height="138" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Algebraproblem.jpg/330px-Algebraproblem.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Algebraproblem.jpg/440px-Algebraproblem.jpg 2x" data-file-width="818" data-file-height="512" /></a><figcaption>A typical algebra problem.</figcaption></figure> <p>The following sections lay out examples of some of the types of algebraic equations that may be encountered. </p> <div class="mw-heading mw-heading3"><h3 id="Linear_equations_with_one_variable">Linear equations with one variable</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=12" title="Edit section: Linear equations with one variable"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Linear_equation" title="Linear equation">Linear equation</a></div> <p>Linear equations are so-called, because when they are plotted, they describe a straight line. The simplest equations to solve are <a href="/wiki/Linear_equation" title="Linear equation">linear equations</a> that have only one variable. They contain only constant numbers and a single variable without an exponent. As an example, consider: </p> <dl><dd>Problem in words: If you double the age of a child and add 4, the resulting answer is 12. How old is the child?</dd></dl> <dl><dd>Equivalent equation: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x+4=12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>4</mn> <mo>=</mo> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x+4=12}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/346c990e543c1b807d25efad677f17a910c44f22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.918ex; height:2.343ex;" alt="{\displaystyle 2x+4=12}"></span> where <span class="texhtml mvar" style="font-style:italic;">x</span> represent the child's age</dd></dl> <p>To solve this kind of equation, the technique is add, subtract, multiply, or divide both sides of the equation by the same number in order to isolate the variable on one side of the equation. Once the variable is isolated, the other side of the equation is the value of the variable.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> This problem and its solution are as follows: </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Divide_large.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Divide_large.gif/220px-Divide_large.gif" decoding="async" width="220" height="207" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/5/59/Divide_large.gif 1.5x" data-file-width="232" data-file-height="218" /></a><figcaption>Solving for x</figcaption></figure> <table> <tbody><tr> <td>1. Equation to solve: </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x+4=12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>4</mn> <mo>=</mo> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x+4=12}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/346c990e543c1b807d25efad677f17a910c44f22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.918ex; height:2.343ex;" alt="{\displaystyle 2x+4=12}"></span> </td></tr> <tr> <td>2. Subtract 4 from both sides: </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x+4-4=12-4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>4</mn> <mo>−</mo> <mn>4</mn> <mo>=</mo> <mn>12</mn> <mo>−</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x+4-4=12-4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1254b15209bd6094b209d4615078d94051bc7ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.924ex; height:2.343ex;" alt="{\displaystyle 2x+4-4=12-4}"></span> </td></tr> <tr> <td>3. This simplifies to: </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x=8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x=8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc58ebcf67c4f2be2a0ad26b646f576dd1617674" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.753ex; height:2.176ex;" alt="{\displaystyle 2x=8}"></span> </td></tr> <tr> <td>4. Divide both sides by 2: </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2x}{2}}={\frac {8}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2x}{2}}={\frac {8}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bbec859078a96fde4ef8382044974dd75c20ace" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.425ex; height:5.176ex;" alt="{\displaystyle {\frac {2x}{2}}={\frac {8}{2}}}"></span> </td></tr> <tr> <td>5. This simplifies to the solution: </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73bfdd8100b6a4ce07d011900560f102e3965064" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=4}"></span> </td></tr></tbody></table> <p>In words: the child is 4 years old. </p><p>The general form of a linear equation with one variable, can be written as: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax+b=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax+b=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08bf93625b100c9a4838fb52ddb9e65acfdb1234" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.503ex; height:2.343ex;" alt="{\displaystyle ax+b=c}"></span> </p><p>Following the same procedure (i.e. subtract <span class="texhtml mvar" style="font-style:italic;">b</span> from both sides, and then divide by <span class="texhtml mvar" style="font-style:italic;">a</span>), the general solution is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\frac {c-b}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>c</mi> <mo>−</mo> <mi>b</mi> </mrow> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {c-b}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae8f1e0d24f24022000a106a0088ca9e9e6db538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.109ex; height:5.343ex;" alt="{\displaystyle x={\frac {c-b}{a}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Linear_equations_with_two_variables">Linear equations with two variables</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=13" title="Edit section: Linear equations with two variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Linear-equations-two-unknowns.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Linear-equations-two-unknowns.svg/220px-Linear-equations-two-unknowns.svg.png" decoding="async" width="220" height="325" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Linear-equations-two-unknowns.svg/330px-Linear-equations-two-unknowns.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Linear-equations-two-unknowns.svg/440px-Linear-equations-two-unknowns.svg.png 2x" data-file-width="469" data-file-height="692" /></a><figcaption>Solving two linear equations with a unique solution at the point that they intersect.</figcaption></figure> <p>A linear equation with two variables has many (i.e. an infinite number of) solutions.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> For example: </p> <dl><dd>Problem in words: A father is 22 years older than his son. How old are they?</dd> <dd>Equivalent equation: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x+22}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mn>22</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x+22}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bafcee05572b20028e3e7c8a97309e21ae5d065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.749ex; height:2.509ex;" alt="{\displaystyle y=x+22}"></span> where <span class="texhtml mvar" style="font-style:italic;">y</span> is the father's age, <span class="texhtml mvar" style="font-style:italic;">x</span> is the son's age.</dd></dl> <p>That cannot be worked out by itself. If the son's age was made known, then there would no longer be two unknowns (variables). The problem then becomes a linear equation with just one variable, that can be solved as described above. </p><p>To solve a linear equation with two variables (unknowns), requires two related equations. For example, if it was also revealed that: </p> <dl><dt>Problem in words</dt> <dd>In 10 years, the father will be twice as old as his son.</dd> <dt>Equivalent equation</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}y+10&=2\times (x+10)\\y&=2\times (x+10)-10&&{\text{Subtract 10 from both sides}}\\y&=2x+20-10&&{\text{Multiple out brackets}}\\y&=2x+10&&{\text{Simplify}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>y</mi> <mo>+</mo> <mn>10</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mo>×</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>10</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mo>×</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>10</mn> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Subtract 10 from both sides</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>20</mn> <mo>−</mo> <mn>10</mn> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Multiple out brackets</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>10</mn> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Simplify</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y+10&=2\times (x+10)\\y&=2\times (x+10)-10&&{\text{Subtract 10 from both sides}}\\y&=2x+20-10&&{\text{Multiple out brackets}}\\y&=2x+10&&{\text{Simplify}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/226b84ad286a2ca4cbf38661fb5ba5d3ed330136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:59.907ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}y+10&=2\times (x+10)\\y&=2\times (x+10)-10&&{\text{Subtract 10 from both sides}}\\y&=2x+20-10&&{\text{Multiple out brackets}}\\y&=2x+10&&{\text{Simplify}}\end{aligned}}}"></span></dd></dl> <p>Now there are two related linear equations, each with two unknowns, which enables the production of a linear equation with just one variable, by subtracting one from the other (called the elimination method):<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}y=x+22&{\text{First equation}}\\y=2x+10&{\text{Second equation}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mn>22</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>First equation</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>10</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Second equation</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}y=x+22&{\text{First equation}}\\y=2x+10&{\text{Second equation}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96611f523b65be48c16243dea368b60b0c67c5b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.135ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}y=x+22&{\text{First equation}}\\y=2x+10&{\text{Second equation}}\end{cases}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&&&{\text{Subtract the first equation from}}\\(y-y)&=(2x-x)+10-22&&{\text{the second in order to remove }}y\\0&=x-12&&{\text{Simplify}}\\12&=x&&{\text{Add 12 to both sides}}\\x&=12&&{\text{Rearrange}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd /> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Subtract the first equation from</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>10</mn> <mo>−</mo> <mn>22</mn> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>the second in order to remove </mtext> </mrow> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mn>12</mn> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Simplify</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>12</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Add 12 to both sides</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>12</mn> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>Rearrange</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&&&{\text{Subtract the first equation from}}\\(y-y)&=(2x-x)+10-22&&{\text{the second in order to remove }}y\\0&=x-12&&{\text{Simplify}}\\12&=x&&{\text{Add 12 to both sides}}\\x&=12&&{\text{Rearrange}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29661b3951958012cea8e8b67f8e16580bed5d05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:66.138ex; height:15.176ex;" alt="{\displaystyle {\begin{aligned}&&&{\text{Subtract the first equation from}}\\(y-y)&=(2x-x)+10-22&&{\text{the second in order to remove }}y\\0&=x-12&&{\text{Simplify}}\\12&=x&&{\text{Add 12 to both sides}}\\x&=12&&{\text{Rearrange}}\end{aligned}}}"></span></dd></dl> <p>In other words, the son is aged 12, and since the father 22 years older, he must be 34. In 10 years, the son will be 22, and the father will be twice his age, 44. This problem is illustrated on the associated plot of the equations. </p><p>For other ways to solve this kind of equations, see below, <b><a href="#System_of_linear_equations">System of linear equations</a></b>. </p> <div class="mw-heading mw-heading3"><h3 id="Quadratic_equations">Quadratic equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=14" title="Edit section: Quadratic equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Quadratic_equation" title="Quadratic equation">Quadratic equation</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Quadratic-equation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Quadratic-equation.svg/220px-Quadratic-equation.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Quadratic-equation.svg/330px-Quadratic-equation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Quadratic-equation.svg/440px-Quadratic-equation.svg.png 2x" data-file-width="570" data-file-height="569" /></a><figcaption>Quadratic equation plot of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x^{2}+3x-10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>−</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x^{2}+3x-10}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/542a39aa589dcbb42f429ee7fd21563e68fd2011" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.136ex; height:3.009ex;" alt="{\displaystyle y=x^{2}+3x-10}"></span> showing its roots at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=-5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=-5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0190f8b44849ea92222977907c26e938566b98a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.399ex; height:2.343ex;" alt="{\displaystyle x=-5}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f39b6e42e5ffb81ac7b051b9e48b9a91d0713c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=2}"></span>, and that the quadratic can be rewritten as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=(x+5)(x-2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=(x+5)(x-2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98e3cbe17adefaec352d99657fbd6bbe6f02ea3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.538ex; height:2.843ex;" alt="{\displaystyle y=(x+5)(x-2)}"></span> </figcaption></figure> <p>A quadratic equation is one which includes a term with an exponent of 2, for example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}"></span>,<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> and no term with higher exponent. The name derives from the Latin <i>quadrus</i>, meaning square.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> In general, a quadratic equation can be expressed in the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax^{2}+bx+c=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax^{2}+bx+c=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23e70cfa003f402d108ec04d97983fb62f69536e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.89ex; height:2.843ex;" alt="{\displaystyle ax^{2}+bx+c=0}"></span>,<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> where <span class="texhtml mvar" style="font-style:italic;">a</span> is not zero (if it were zero, then the equation would not be quadratic but linear). Because of this a quadratic equation must contain the term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1a63fb574ed624044abbe6aeebcb600d0cb9802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.614ex; height:2.676ex;" alt="{\displaystyle ax^{2}}"></span>, which is known as the quadratic term. Hence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f455a7f96d74aa94573d8e32da3b240ab0aa294f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.491ex; height:2.676ex;" alt="{\displaystyle a\neq 0}"></span>, and so we may divide by <span class="texhtml mvar" style="font-style:italic;">a</span> and rearrange the equation into the standard form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+px+q=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>p</mi> <mi>x</mi> <mo>+</mo> <mi>q</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+px+q=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1447b7f80b86966dc9f431ba415b2de2608e50e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.894ex; height:3.009ex;" alt="{\displaystyle x^{2}+px+q=0}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p={\frac {b}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\frac {b}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4863d307e928bab8d79085004f614d545a010b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:6.423ex; height:5.343ex;" alt="{\displaystyle p={\frac {b}{a}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q={\frac {c}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q={\frac {c}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35385a31b58ac8828c8bb9c9f92e6e9ec5372ef8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.234ex; height:4.676ex;" alt="{\displaystyle q={\frac {c}{a}}}"></span>. Solving this, by a process known as <a href="/wiki/Completing_the_square" title="Completing the square">completing the square</a>, leads to the <a href="/wiki/Quadratic_formula" title="Quadratic formula">quadratic formula</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>b</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mi>c</mi> </msqrt> </mrow> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dca3ce631f83b33ad881aed5f6e12e2f6c3afbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.172ex; height:6.176ex;" alt="{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}},}"></span></dd></dl> <p>where <a href="/wiki/Plus%E2%80%93minus_sign" title="Plus–minus sign">the symbol "±"</a> indicates that both </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\quad {\text{and}}\quad x={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>b</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mi>c</mi> </msqrt> </mrow> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>b</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mi>c</mi> </msqrt> </mrow> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\quad {\text{and}}\quad x={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d03b5b5af5ed622791d827b7e3661efb8a3db1f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:51.443ex; height:6.176ex;" alt="{\displaystyle x={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\quad {\text{and}}\quad x={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}}"></span></dd></dl> <p>are solutions of the quadratic equation. </p><p>Quadratic equations can also be solved using <a href="/wiki/Factorization" title="Factorization">factorization</a> (the reverse process of which is <a href="/wiki/Polynomial_expansion" title="Polynomial expansion">expansion</a>, but for two <a href="/wiki/Linear_function" title="Linear function">linear terms</a> is sometimes denoted <a href="/wiki/FOIL_rule" class="mw-redirect" title="FOIL rule">foiling</a>). As an example of factoring: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+3x-10=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>−</mo> <mn>10</mn> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+3x-10=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21d195781d3bba5b1c3a35b2790f73fe42f9f637" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.79ex; height:3.009ex;" alt="{\displaystyle x^{2}+3x-10=0,}"></span></dd></dl> <p>which is the same thing as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+5)(x-2)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+5)(x-2)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01091b269645322d27a8f5b12cc54060862707cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.191ex; height:2.843ex;" alt="{\displaystyle (x+5)(x-2)=0.}"></span></dd></dl> <p>It follows from the <a href="/wiki/Zero-product_property" title="Zero-product property">zero-product property</a> that either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f39b6e42e5ffb81ac7b051b9e48b9a91d0713c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=2}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=-5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=-5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0190f8b44849ea92222977907c26e938566b98a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.399ex; height:2.343ex;" alt="{\displaystyle x=-5}"></span> are the solutions, since precisely one of the factors must be equal to <a href="/wiki/Zero" class="mw-redirect" title="Zero">zero</a>. All quadratic equations will have two solutions in the <a href="/wiki/Complex_number" title="Complex number">complex number</a> system, but need not have any in the <a href="/wiki/Real_number" title="Real number">real number</a> system. For example, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+1=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e01c67127b28bb80e2102c934d0d01daa5c20a61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.648ex; height:2.843ex;" alt="{\displaystyle x^{2}+1=0}"></span></dd></dl> <p>has no real number solution since no real number squared equals −1. Sometimes a quadratic equation has a root of <a href="/wiki/Multiplicity_(mathematics)" title="Multiplicity (mathematics)">multiplicity</a> 2, such as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+1)^{2}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+1)^{2}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b839672eac67f8ac41cabb303c08b87f802869a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.104ex; height:3.176ex;" alt="{\displaystyle (x+1)^{2}=0.}"></span></dd></dl> <p>For this equation, −1 is a root of multiplicity 2. This means −1 appears twice, since the equation can be rewritten in factored form as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x-(-1)][x-(-1)]=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo stretchy="false">[</mo> <mi>x</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x-(-1)][x-(-1)]=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ea674f8b3cef9f4350a8ce21f48fc09d99cb539" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.395ex; height:2.843ex;" alt="{\displaystyle [x-(-1)][x-(-1)]=0.}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Complex_numbers">Complex numbers</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=15" title="Edit section: Complex numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>All quadratic equations have exactly two solutions in <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> (but they may be equal to each other), a category that includes <a href="/wiki/Real_number" title="Real number">real numbers</a>, <a href="/wiki/Imaginary_number" title="Imaginary number">imaginary numbers</a>, and sums of real and imaginary numbers. Complex numbers first arise in the teaching of quadratic equations and the quadratic formula. For example, the quadratic equation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+x+1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+x+1=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c5e55ad884c42814df1c0e098670f34d685d0eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.818ex; height:2.843ex;" alt="{\displaystyle x^{2}+x+1=0}"></span></dd></dl> <p>has solutions </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\frac {-1+{\sqrt {-3}}}{2}}\quad \quad {\text{and}}\quad \quad x={\frac {-1-{\sqrt {-3}}}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>3</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="1em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mspace width="1em" /> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>3</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {-1+{\sqrt {-3}}}{2}}\quad \quad {\text{and}}\quad \quad x={\frac {-1-{\sqrt {-3}}}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd41f4fbb95aaf7dcc832f376d228077f3874ba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:45.648ex; height:5.843ex;" alt="{\displaystyle x={\frac {-1+{\sqrt {-3}}}{2}}\quad \quad {\text{and}}\quad \quad x={\frac {-1-{\sqrt {-3}}}{2}}.}"></span></dd></dl> <p>Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49e01b99dd42d59d290fdef7722b870efe61eb2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-3}}}"></span> is not any real number, both of these solutions for <i>x</i> are complex numbers. </p> <div class="mw-heading mw-heading3"><h3 id="Exponential_and_logarithmic_equations">Exponential and logarithmic equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=16" title="Edit section: Exponential and logarithmic equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Logarithm" title="Logarithm">Logarithm</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Binary_logarithm_plot_with_ticks.svg" class="mw-file-description"><img alt="Graph showing a logarithm curves, which crosses the x-axis where x is 1 and extend towards minus infinity along the y-axis." src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Binary_logarithm_plot_with_ticks.svg/300px-Binary_logarithm_plot_with_ticks.svg.png" decoding="async" width="300" height="239" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Binary_logarithm_plot_with_ticks.svg/450px-Binary_logarithm_plot_with_ticks.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Binary_logarithm_plot_with_ticks.svg/600px-Binary_logarithm_plot_with_ticks.svg.png 2x" data-file-width="408" data-file-height="325" /></a><figcaption>The <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> of the logarithm to base 2 crosses the <a href="/wiki/X_axis" class="mw-redirect" title="X axis"><i>x</i> axis</a> (horizontal axis) at 1 and passes through the points with <a href="/wiki/Coordinate" class="mw-redirect" title="Coordinate">coordinates</a> <span class="nowrap">(2, 1)</span>, <span class="nowrap">(4, 2)</span>, and <span class="nowrap">(8, 3)</span>. For example, <span class="nowrap">log<sub>2</sub>(8) = 3</span>, because <span class="nowrap">2<sup>3</sup> = 8.</span> The graph gets arbitrarily close to the <i>y</i> axis, but <a href="/wiki/Asymptotic" class="mw-redirect" title="Asymptotic">does not meet or intersect it</a>.</figcaption></figure> <p>An exponential equation is one which has the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{x}=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{x}=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/009ac5470a4ecd0afab8866426ad920c494316c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.498ex; height:2.343ex;" alt="{\displaystyle a^{x}=b}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34a80ea013edb56e340b19550430a8b6dfd7b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a>0}"></span>,<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> which has solution </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\log _{a}b={\frac {\ln b}{\ln a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>b</mi> </mrow> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>a</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\log _{a}b={\frac {\ln b}{\ln a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39ff639098e9d111ac517625feb528637c4ce8a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.377ex; height:5.509ex;" alt="{\displaystyle x=\log _{a}b={\frac {\ln b}{\ln a}}}"></span></dd></dl> <p>when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94436473a90bd55191a79c59474cb5456dcbec00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b>0}"></span>. Elementary algebraic techniques are used to rewrite a given equation in the above way before arriving at the solution. For example, if </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\cdot 2^{x-1}+1=10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\cdot 2^{x-1}+1=10}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d77071814152240876f6c13181b8738ecb611e3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.703ex; height:2.843ex;" alt="{\displaystyle 3\cdot 2^{x-1}+1=10}"></span></dd></dl> <p>then, by subtracting 1 from both sides of the equation, and then dividing both sides by 3 we obtain </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{x-1}=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{x-1}=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ac896aac2850d21edf38fbda9988efd7cb16469" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.696ex; height:2.676ex;" alt="{\displaystyle 2^{x-1}=3}"></span></dd></dl> <p>whence </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-1=\log _{2}3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-1=\log _{2}3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96ccb53695d9611a82433817e2405e6ed7df35c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.007ex; height:2.676ex;" alt="{\displaystyle x-1=\log _{2}3}"></span></dd></dl> <p>or </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\log _{2}3+1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mn>3</mn> <mo>+</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\log _{2}3+1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9720b577b6c98343f0032293a3dcb5190174e7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.653ex; height:2.676ex;" alt="{\displaystyle x=\log _{2}3+1.}"></span></dd></dl> <p>A logarithmic equation is an equation of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle log_{a}(x)=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mi>o</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle log_{a}(x)=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f0b748099342737433ad3523aae1afdf87ac82e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.267ex; height:2.843ex;" alt="{\displaystyle log_{a}(x)=b}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34a80ea013edb56e340b19550430a8b6dfd7b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a>0}"></span>, which has solution </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a^{b}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=a^{b}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61206cf1172d089e925f3bd1aba8789a73fef7c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.242ex; height:2.676ex;" alt="{\displaystyle x=a^{b}.}"></span></dd></dl> <p>For example, if </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\log _{5}(x-3)-2=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\log _{5}(x-3)-2=6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48756f6a38589bbfc6e5e3ccc614db69447da988" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.981ex; height:2.843ex;" alt="{\displaystyle 4\log _{5}(x-3)-2=6}"></span></dd></dl> <p>then, by adding 2 to both sides of the equation, followed by dividing both sides by 4, we get </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{5}(x-3)=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{5}(x-3)=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4a77c02b98ff14bed821b26f955e1cd545ceb4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.429ex; height:2.843ex;" alt="{\displaystyle \log _{5}(x-3)=2}"></span></dd></dl> <p>whence </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-3=5^{2}=25}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mo>=</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>25</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-3=5^{2}=25}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57a64460e334e40242f98165c068c36d2eab5dd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.071ex; height:2.843ex;" alt="{\displaystyle x-3=5^{2}=25}"></span></dd></dl> <p>from which we obtain </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=28.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>28.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=28.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c829eab2e5b164e590a5bfeb4f83b0ea3a7fcbce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.4ex; height:2.176ex;" alt="{\displaystyle x=28.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Radical_equations">Radical equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=17" title="Edit section: Radical equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="thumb tright" style=""><div class="thumbinner" style="width:152px"><div class="thumbimage noresize" style="width:150px;"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overset {}{\underset {}{{\sqrt[{2}]{x^{3}}}\equiv x^{\frac {3}{2}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <munder> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mroot> </mrow> <mo>≡</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mrow /> </munder> <mrow /> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overset {}{\underset {}{{\sqrt[{2}]{x^{3}}}\equiv x^{\frac {3}{2}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/611dbd1edaec2b3d704377cd9d9e299e9fdd1ddd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.872ex; height:6.176ex;" alt="{\displaystyle {\overset {}{\underset {}{{\sqrt[{2}]{x^{3}}}\equiv x^{\frac {3}{2}}}}}}"></span></div><div class="thumbcaption">Radical equation showing two ways to represent the same expression. The triple bar means the equation is true for all values of <i>x</i></div></div></div> <p>A radical equation is one that includes a radical sign, which includes <a href="/wiki/Square_root" title="Square root">square roots</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9763d9ff8638b882608213f6dc8e9fc8ddc79d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.912ex; height:3.009ex;" alt="{\displaystyle {\sqrt {x}},}"></span> <a href="/wiki/Cube_root" title="Cube root">cube roots</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{3}]{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a55f866116e7a86823816615dd98fcccde75473" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.266ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{3}]{x}}}"></span>, and <a href="/wiki/Nth_root" title="Nth root"><i>n</i>th roots</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b3ba2638d05cd9ed8dafae7e34986399e48ea99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.266ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{x}}}"></span>. Recall that an <i>n</i>th root can be rewritten in exponential format, so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b3ba2638d05cd9ed8dafae7e34986399e48ea99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.266ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{x}}}"></span> is equivalent to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\frac {1}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\frac {1}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55c86c68cf18b5f2f38c440bee62562ff4f304f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.199ex; height:3.343ex;" alt="{\displaystyle x^{\frac {1}{n}}}"></span>. Combined with regular exponents (powers), then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{2}]{x^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{2}]{x^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6689fad02ba04851cff57ef80164ad8b1049f847" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.708ex; height:3.343ex;" alt="{\displaystyle {\sqrt[{2}]{x^{3}}}}"></span> (the square root of <span class="texhtml mvar" style="font-style:italic;">x</span> cubed), can be rewritten as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\frac {3}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\frac {3}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a19ab17bfacbb7976f7f90799c9fd79241d5433" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.065ex; height:3.509ex;" alt="{\displaystyle x^{\frac {3}{2}}}"></span>.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> So a common form of a radical equation is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{x^{m}}}=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{x^{m}}}=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c088e9f58767bab398f26af86cca6ec2ff0fde1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.269ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{n}]{x^{m}}}=a}"></span> (equivalent to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\frac {m}{n}}=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mrow> </msup> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\frac {m}{n}}=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b63addf2cbcdd590d1c725337bffa31f8edc439" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.898ex; height:3.009ex;" alt="{\displaystyle x^{\frac {m}{n}}=a}"></span>) where <span class="texhtml mvar" style="font-style:italic;">m</span> and <span class="texhtml mvar" style="font-style:italic;">n</span> are <a href="/wiki/Integer" title="Integer">integers</a>. It has <a href="/wiki/Real_number" title="Real number">real</a> solution(s): </p> <table class="wikitable" style="text-align:center"> <tbody><tr style="vertical-align:top"> <th><span class="texhtml mvar" style="font-style:italic;">n</span> is odd </th> <th><span class="texhtml mvar" style="font-style:italic;">n</span> is even<br />and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c33f2f7ac8b81e1296d581427bdb2d30a50a2b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.491ex; height:2.343ex;" alt="{\displaystyle a\geq 0}"></span> </th> <th><span class="texhtml mvar" style="font-style:italic;">n</span> <b>and</b> <span class="texhtml mvar" style="font-style:italic;">m</span> are <b>even</b><br /><b>and</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d7ca60f6ed64b99649dcee21847295fedf206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a<0}"></span> </th> <th><span class="texhtml mvar" style="font-style:italic;">n</span> <b>is even</b>, <span class="texhtml mvar" style="font-style:italic;">m</span> <b>is odd</b>, <b>and</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d7ca60f6ed64b99649dcee21847295fedf206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a<0}"></span> </th></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\sqrt[{n}]{a^{m}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\sqrt[{n}]{a^{m}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf36c992d5c1f85c575576d9589578164553e181" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.269ex; height:3.009ex;" alt="{\displaystyle x={\sqrt[{n}]{a^{m}}}}"></span> <p>equivalently </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\left({\sqrt[{n}]{a}}\right)^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\left({\sqrt[{n}]{a}}\right)^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d05f4804fdd0334e43ab0660a0028909164ccbc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.078ex; height:3.176ex;" alt="{\displaystyle x=\left({\sqrt[{n}]{a}}\right)^{m}}"></span></dd></dl> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\pm {\sqrt[{n}]{a^{m}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\pm {\sqrt[{n}]{a^{m}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/683d049337b497ee6f0afd2913dffb359495ab44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.077ex; height:3.009ex;" alt="{\displaystyle x=\pm {\sqrt[{n}]{a^{m}}}}"></span> <p>equivalently </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\pm \left({\sqrt[{n}]{a}}\right)^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>±</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\pm \left({\sqrt[{n}]{a}}\right)^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71b48d8b76129457136e984c5a34c48c366a85dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.886ex; height:3.176ex;" alt="{\displaystyle x=\pm \left({\sqrt[{n}]{a}}\right)^{m}}"></span></dd></dl> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\pm {\sqrt[{n}]{a^{m}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\pm {\sqrt[{n}]{a^{m}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/683d049337b497ee6f0afd2913dffb359495ab44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.077ex; height:3.009ex;" alt="{\displaystyle x=\pm {\sqrt[{n}]{a^{m}}}}"></span> </td> <td>no real solution </td></tr></tbody></table> <p>For example, if: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+5)^{2/3}=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>5</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+5)^{2/3}=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa025dc7ac95370091a5046a6f9e85703fd9297a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.101ex; height:3.343ex;" alt="{\displaystyle (x+5)^{2/3}=4}"></span></dd></dl> <p>then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x+5&=\pm ({\sqrt {4}})^{3},\\x+5&=\pm 8,\\x&=-5\pm 8,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> <mo>+</mo> <mn>5</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>±</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>4</mn> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> <mo>+</mo> <mn>5</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>±</mo> <mn>8</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>5</mn> <mo>±</mo> <mn>8</mn> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x+5&=\pm ({\sqrt {4}})^{3},\\x+5&=\pm 8,\\x&=-5\pm 8,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c2fbd2d1fd5da9dceb009dbafdb3113bf1ea750" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:17.599ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}x+5&=\pm ({\sqrt {4}})^{3},\\x+5&=\pm 8,\\x&=-5\pm 8,\end{aligned}}}"></span></dd></dl> <p>and thus </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=3\quad {\text{or}}\quad x=-13}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>3</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>or</mtext> </mrow> <mspace width="1em" /> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>13</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=3\quad {\text{or}}\quad x=-13}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/460bd3d2a4b4640676187faaeeb05892f6e54323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.871ex; height:2.343ex;" alt="{\displaystyle x=3\quad {\text{or}}\quad x=-13}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="System_of_linear_equations">System of linear equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=18" title="Edit section: System of linear equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/System_of_linear_equations" title="System of linear equations">System of linear equations</a></div> <p>There are different methods to solve a system of linear equations with two variables. </p> <div class="mw-heading mw-heading4"><h4 id="Elimination_method">Elimination method</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=19" title="Edit section: Elimination method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Intersecting_Lines.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Intersecting_Lines.svg/220px-Intersecting_Lines.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Intersecting_Lines.svg/330px-Intersecting_Lines.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Intersecting_Lines.svg/440px-Intersecting_Lines.svg.png 2x" data-file-width="500" data-file-height="500" /></a><figcaption>The solution set for the equations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-y=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-y=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/107763e42385143d3d69090af16c1f45c066d49a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.395ex; height:2.509ex;" alt="{\displaystyle x-y=-1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3x+y=9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>=</mo> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x+y=9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/047b72c6cc8d183ab9fedb331ff24d76d6703c52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.749ex; height:2.509ex;" alt="{\displaystyle 3x+y=9}"></span> is the single point (2, 3).</figcaption></figure> <p>An example of solving a system of linear equations is by using the elimination method: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}4x+2y&=14\\2x-y&=1.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>y</mi> </mtd> <mtd> <mo>=</mo> <mn>14</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mtd> <mtd> <mo>=</mo> <mn>1.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}4x+2y&=14\\2x-y&=1.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f2d87cc6ac7e78a26fdc02b68941dc01dc41b55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.246ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}4x+2y&=14\\2x-y&=1.\end{cases}}}"></span></dd></dl> <p>Multiplying the terms in the second equation by 2: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4x+2y=14}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>y</mi> <mo>=</mo> <mn>14</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4x+2y=14}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f808b7f9403c7ed113e148457240a3fea14726b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.074ex; height:2.509ex;" alt="{\displaystyle 4x+2y=14}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4x-2y=2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo>=</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4x-2y=2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b5e83a51011ff3cf00285e3892383560f609410" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.558ex; height:2.509ex;" alt="{\displaystyle 4x-2y=2.}"></span></dd></dl> <p>Adding the two equations together to get: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8x=16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mi>x</mi> <mo>=</mo> <mn>16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8x=16}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2643249fdfb0c7f1c4d5d4125335be274cdc073" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.916ex; height:2.176ex;" alt="{\displaystyle 8x=16}"></span></dd></dl> <p>which simplifies to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ba0d302dee657b740f239df7d781071f6c247b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.237ex; height:2.176ex;" alt="{\displaystyle x=2.}"></span></dd></dl> <p>Since the fact that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f39b6e42e5ffb81ac7b051b9e48b9a91d0713c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=2}"></span> is known, it is then possible to deduce that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6284ea4bb2d82b7a988082dd286adbb9dd095356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y=3}"></span> by either of the original two equations (by using <i>2</i> instead of <span class="texhtml mvar" style="font-style:italic;">x</span> ) The full solution to this problem is then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}x=2\\y=3.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>x</mi> <mo>=</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo>=</mo> <mn>3.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}x=2\\y=3.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4bd4bc969d06560b090bcd79ec60c912d368d6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.558ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}x=2\\y=3.\end{cases}}}"></span></dd></dl> <p>This is not the only way to solve this specific system; <span class="texhtml mvar" style="font-style:italic;">y</span> could have been resolved before <span class="texhtml mvar" style="font-style:italic;">x</span>. </p> <div class="mw-heading mw-heading4"><h4 id="Substitution_method">Substitution method</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=20" title="Edit section: Substitution method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another way of solving the same system of linear equations is by substitution. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}4x+2y&=14\\2x-y&=1.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>y</mi> </mtd> <mtd> <mo>=</mo> <mn>14</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mtd> <mtd> <mo>=</mo> <mn>1.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}4x+2y&=14\\2x-y&=1.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f2d87cc6ac7e78a26fdc02b68941dc01dc41b55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.246ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}4x+2y&=14\\2x-y&=1.\end{cases}}}"></span></dd></dl> <p>An equivalent for <span class="texhtml mvar" style="font-style:italic;">y</span> can be deduced by using one of the two equations. Using the second equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x-y=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x-y=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e08b49c74c2109c877dfd8893ee2e12ef4227b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.749ex; height:2.509ex;" alt="{\displaystyle 2x-y=1}"></span></dd></dl> <p>Subtracting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e50b849d3a7cd902f0ae3fa6ad6d1cad49987c39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.492ex; height:2.176ex;" alt="{\displaystyle 2x}"></span> from each side of the equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}2x-2x-y&=1-2x\\-y&=1-2x\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}2x-2x-y&=1-2x\\-y&=1-2x\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/188d10b5e0e191e0d83eefa48e3c030960c3f864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.165ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}2x-2x-y&=1-2x\\-y&=1-2x\end{aligned}}}"></span></dd></dl> <p>and multiplying by −1: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=2x-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=2x-1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65064963b28ae18d96fa2e4218a97cc176fff7e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.396ex; height:2.509ex;" alt="{\displaystyle y=2x-1.}"></span></dd></dl> <p>Using this <span class="texhtml mvar" style="font-style:italic;">y</span> value in the first equation in the original system: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}4x+2(2x-1)&=14\\4x+4x-2&=14\\8x-2&=14\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>14</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>14</mn> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>14</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}4x+2(2x-1)&=14\\4x+4x-2&=14\\8x-2&=14\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b204003cd2f3139da95931466a361534cdf5026a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:20.974ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}4x+2(2x-1)&=14\\4x+4x-2&=14\\8x-2&=14\end{aligned}}}"></span></dd></dl> <p>Adding <i>2</i> on each side of the equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}8x-2+2&=14+2\\8x&=16\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>8</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo>+</mo> <mn>2</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>14</mn> <mo>+</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>16</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}8x-2+2&=14+2\\8x&=16\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47382f34735f0fc0d4923ec1496ca70c8c930311" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.676ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}8x-2+2&=14+2\\8x&=16\end{aligned}}}"></span></dd></dl> <p>which simplifies to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f39b6e42e5ffb81ac7b051b9e48b9a91d0713c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=2}"></span></dd></dl> <p>Using this value in one of the equations, the same solution as in the previous method is obtained. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}x=2\\y=3.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>x</mi> <mo>=</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo>=</mo> <mn>3.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}x=2\\y=3.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4bd4bc969d06560b090bcd79ec60c912d368d6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.558ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}x=2\\y=3.\end{cases}}}"></span></dd></dl> <p>This is not the only way to solve this specific system; in this case as well, <span class="texhtml mvar" style="font-style:italic;">y</span> could have been solved before <span class="texhtml mvar" style="font-style:italic;">x</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Other_types_of_systems_of_linear_equations">Other types of systems of linear equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=21" title="Edit section: Other types of systems of linear equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Inconsistent_systems">Inconsistent systems</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=22" title="Edit section: Inconsistent systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Parallel_Lines.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Parallel_Lines.svg/220px-Parallel_Lines.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Parallel_Lines.svg/330px-Parallel_Lines.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/Parallel_Lines.svg/440px-Parallel_Lines.svg.png 2x" data-file-width="1089" data-file-height="1089" /></a><figcaption>The equations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3x+2y=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>y</mi> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x+2y=6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c069a4331c966648337f5ea1db220afce23514d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.911ex; height:2.509ex;" alt="{\displaystyle 3x+2y=6}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3x+2y=12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>y</mi> <mo>=</mo> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x+2y=12}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58c127ab08640639903e2d5e1412709d6a7d35c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.074ex; height:2.509ex;" alt="{\displaystyle 3x+2y=12}"></span> are parallel and cannot intersect, and is unsolvable.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Quadratic-linear-equations.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Quadratic-linear-equations.svg/220px-Quadratic-linear-equations.svg.png" decoding="async" width="220" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Quadratic-linear-equations.svg/330px-Quadratic-linear-equations.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Quadratic-linear-equations.svg/440px-Quadratic-linear-equations.svg.png 2x" data-file-width="572" data-file-height="569" /></a><figcaption>Plot of a quadratic equation (red) and a linear equation (blue) that do not intersect, and consequently for which there is no common solution.</figcaption></figure> <p>In the above example, a solution exists. However, there are also systems of equations which do not have any solution. Such a system is called <a href="/wiki/Inconsistent_system" class="mw-redirect" title="Inconsistent system">inconsistent</a>. An obvious example is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\begin{aligned}x+y&=1\\0x+0y&=2\,.\end{aligned}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mi>x</mi> <mo>+</mo> <mn>0</mn> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\begin{aligned}x+y&=1\\0x+0y&=2\,.\end{aligned}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d16b7301ea325d3bed4a98ce404bf5c12f00b964" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.191ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}{\begin{aligned}x+y&=1\\0x+0y&=2\,.\end{aligned}}\end{cases}}}"></span></dd></dl> <p>As 0≠2, the second equation in the system has no solution. Therefore, the system has no solution. However, not all inconsistent systems are recognized at first sight. As an example, consider the system </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\begin{aligned}4x+2y&=12\\-2x-y&=-4\,.\end{aligned}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>12</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>4</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\begin{aligned}4x+2y&=12\\-2x-y&=-4\,.\end{aligned}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bb2eaef955f8943956aa58acdca1719c6d618e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.645ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}{\begin{aligned}4x+2y&=12\\-2x-y&=-4\,.\end{aligned}}\end{cases}}}"></span></dd></dl> <p>Multiplying by 2 both sides of the second equation, and adding it to the first one results in </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0x+0y=4\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mi>x</mi> <mo>+</mo> <mn>0</mn> <mi>y</mi> <mo>=</mo> <mn>4</mn> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0x+0y=4\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19223a6e22bc6207f62f1794a23e336228eccb21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.945ex; height:2.509ex;" alt="{\displaystyle 0x+0y=4\,,}"></span></dd></dl> <p>which clearly has no solution. </p> <div class="mw-heading mw-heading4"><h4 id="Undetermined_systems">Undetermined systems</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=23" title="Edit section: Undetermined systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are also systems which have infinitely many solutions, in contrast to a system with a unique solution (meaning, a unique pair of values for <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span>) For example: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\begin{aligned}4x+2y&=12\\-2x-y&=-6\end{aligned}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>12</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>6</mn> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\begin{aligned}4x+2y&=12\\-2x-y&=-6\end{aligned}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0a7d5f209da787f90421c56fb21b9561b1db753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.611ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}{\begin{aligned}4x+2y&=12\\-2x-y&=-6\end{aligned}}\end{cases}}}"></span></dd></dl> <p>Isolating <span class="texhtml mvar" style="font-style:italic;">y</span> in the second equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=-2x+6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=-2x+6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5b7a0f213eacc838e16f1b74ab4644665cef145" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.557ex; height:2.509ex;" alt="{\displaystyle y=-2x+6}"></span></dd></dl> <p>And using this value in the first equation in the system: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}4x+2(-2x+6)=12\\4x-4x+12=12\\12=12\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>12</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>12</mn> <mo>=</mo> <mn>12</mn> </mtd> </mtr> <mtr> <mtd> <mn>12</mn> <mo>=</mo> <mn>12</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}4x+2(-2x+6)=12\\4x-4x+12=12\\12=12\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62ec343dafcb09463d30f4183660379740d8a19a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:22.782ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}4x+2(-2x+6)=12\\4x-4x+12=12\\12=12\end{aligned}}}"></span></dd></dl> <p>The equality is true, but it does not provide a value for <span class="texhtml mvar" style="font-style:italic;">x</span>. Indeed, one can easily verify (by just filling in some values of <span class="texhtml mvar" style="font-style:italic;">x</span>) that for any <span class="texhtml mvar" style="font-style:italic;">x</span> there is a solution as long as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=-2x+6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=-2x+6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5b7a0f213eacc838e16f1b74ab4644665cef145" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.557ex; height:2.509ex;" alt="{\displaystyle y=-2x+6}"></span>. There is an infinite number of solutions for this system. </p> <div class="mw-heading mw-heading4"><h4 id="Over-_and_underdetermined_systems">Over- and underdetermined systems</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=24" title="Edit section: Over- and underdetermined systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Systems with more variables than the number of linear equations are called <a href="/wiki/Underdetermined_system" title="Underdetermined system">underdetermined</a>. Such a system, if it has any solutions, does not have a unique one but rather an infinitude of them. An example of such a system is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\begin{aligned}x+2y&=10\\y-z&=2.\end{aligned}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>10</mn> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2.</mn> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\begin{aligned}x+2y&=10\\y-z&=2.\end{aligned}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e4ae0ac2282dbd3ee48d3333da4ba22535ca78e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.157ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}{\begin{aligned}x+2y&=10\\y-z&=2.\end{aligned}}\end{cases}}}"></span></dd></dl> <p>When trying to solve it, one is led to express some variables as functions of the other ones if any solutions exist, but cannot express <i>all</i> solutions <a href="/wiki/Number" title="Number">numerically</a> because there are an infinite number of them if there are any. </p><p>A system with a higher number of equations than variables is called <a href="/wiki/Overdetermined_system" title="Overdetermined system">overdetermined</a>. If an overdetermined system has any solutions, necessarily some equations are <a href="/wiki/Linear_combination" title="Linear combination">linear combinations</a> of the others. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=25" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/History_of_algebra" title="History of algebra">History of algebra</a></li> <li><a href="/wiki/Binary_operation" title="Binary operation">Binary operation</a></li> <li><a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a></li> <li><a href="/wiki/Mathematics_education" title="Mathematics education">Mathematics education</a></li> <li><a href="/wiki/Number_line" title="Number line">Number line</a></li> <li><a href="/wiki/Polynomial" title="Polynomial">Polynomial</a></li> <li><a href="/wiki/Cancelling_out" title="Cancelling out">Cancelling out</a></li> <li><a href="/wiki/Tarski%27s_high_school_algebra_problem" title="Tarski's high school algebra problem">Tarski's high school algebra problem</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=26" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, <i> <a href="/wiki/Elements_of_Algebra" title="Elements of Algebra">Elements of Algebra</a></i>, 1770. English translation <a href="/w/index.php?title=Tarquin_Press&action=edit&redlink=1" class="new" title="Tarquin Press (page does not exist)">Tarquin Press</a>, 2007, <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-899618-79-8" title="Special:BookSources/978-1-899618-79-8">978-1-899618-79-8</a>, also online digitized editions<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> 2006,<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> 1822.</li> <li>Charles Smith, <i><a rel="nofollow" class="external text" href="http://digital.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=smit025">A Treatise on Algebra</a></i>, in <a rel="nofollow" class="external text" href="http://historical.library.cornell.edu/math">Cornell University Library Historical Math Monographs</a>.</li> <li>Redden, John. <a rel="nofollow" class="external text" href="http://catalog.flatworldknowledge.com/bookhub/reader/128?e=fwk-redden-ch01"><i>Elementary Algebra</i></a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160610165651/http://catalog.flatworldknowledge.com/bookhub/reader/128?e=fwk-redden-ch01">Archived</a> 2016-06-10 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Flat World Knowledge, 2011</li></ul> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Pierce, R., <a rel="nofollow" class="external text" href="https://www.mathsisfun.com/algebra/index-college.html">College Algebra</a>, <i>Maths is Fun</i>, accessed 28 August 2023</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="/wiki/Herbert_Ellsworth_Slaught" title="Herbert Ellsworth Slaught">H.E. Slaught</a> and N.J. Lennes, <i>Elementary algebra</i>, Publ. Allyn and Bacon, 1915, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gLii_eO4dNsC&dq=%22Elementary%20algebra%22%20letters&pg=PA1">page 1</a> (republished by Forgotten Books)</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Lewis Hirsch, Arthur Goodman, <i>Understanding Elementary Algebra With Geometry: A Course for College Students</i>, Publisher: Cengage Learning, 2005, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0534999727" title="Special:BookSources/0534999727">0534999727</a>, 9780534999728, 654 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7hdK4RSub5cC&q=generalization&pg=PA2">page 2</a></span> </li> <li id="cite_note-leff-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-leff_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-leff_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Lawrence S. Leff, <i>College Algebra: Barron's Ez-101 Study Keys</i>, Publisher: Barron's Educational Series, 2005, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0764129147" title="Special:BookSources/0764129147">0764129147</a>, 9780764129148, 230 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XesryURrNKAC&dq=algebra+variables+generalize&pg=PA2">page 2</a></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.encyclopedia.com/environment/encyclopedias-almanacs-transcripts-and-maps/algebraic-operation">"algebraic operation | Encyclopedia.com"</a>. <i>www.encyclopedia.com</i><span class="reference-accessdate">. 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Retrieved <span class="nowrap">2020-08-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=encyclopediaofmath.org&rft.atitle=Algebraic+operation+-+Encyclopedia+of+Mathematics&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FAlgebraic_operation&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElementary+algebra" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Richard N. Aufmann, Joanne Lockwood, <i>Introductory Algebra: An Applied Approach</i>, Publisher Cengage Learning, 2010, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1439046042" title="Special:BookSources/1439046042">1439046042</a>, 9781439046043, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MPIWikTHVXQC&q=coefficient+&pg=PA78">page 78</a></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">William L. Hosch (editor), <i>The Britannica Guide to Algebra and Trigonometry</i>, Britannica Educational Publishing, The Rosen Publishing Group, 2010, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1615302190" title="Special:BookSources/1615302190">1615302190</a>, 9781615302192, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ad0P0elU1_0C&q=letters&pg=PA71">page 71</a></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">James E. Gentle, <i>Numerical Linear Algebra for Applications in Statistics</i>, Publisher: Springer, 1998, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0387985425" title="Special:BookSources/0387985425">0387985425</a>, 9780387985428, 221 pages, [James E. Gentle page 184]</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Horatio Nelson Robinson, <i>New elementary algebra: containing the rudiments of science for schools and academies</i>, Ivison, Phinney, Blakeman, & Co., 1866, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=dKZXAAAAYAAJ&dq=Elementary+algebra+notation&pg=PA7">page 7</a></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Ron Larson, Robert Hostetler, Bruce H. Edwards, <i>Algebra And Trigonometry: A Graphing Approach</i>, Publisher: Cengage Learning, 2007, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/061885195X" title="Special:BookSources/061885195X">061885195X</a>, 9780618851959, 1114 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5iXVZHhkjAgC&dq=operations+addition%2C+subtraction%2C+multiplication%2C+division+exponentiation.&pg=PA6">page 6</a></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Sin Kwai Meng, Chip Wai Lung, Ng Song Beng, "Algebraic notation", in <i>Mathematics Matters Secondary 1 Express Textbook</i>, Publisher Panpac Education Pte Ltd, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9812738827" title="Special:BookSources/9812738827">9812738827</a>, 9789812738820, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nL5ObMmDvPEC&dq=%22Algebraic+notation%22+multiplication+omitted&pg=PR9-IA8">page 68</a></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">David Alan Herzog, <i>Teach Yourself Visually Algebra</i>, Publisher John Wiley & Sons, 2008, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0470185597" title="Special:BookSources/0470185597">0470185597</a>, 9780470185599, 304 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Igs6t_clf0oC&q=coefficient+of+1&pg=PA72">page 72</a></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">John C. Peterson, <i>Technical Mathematics With Calculus</i>, Publisher Cengage Learning, 2003, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0766861899" title="Special:BookSources/0766861899">0766861899</a>, 9780766861893, 1613 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PGuSDjHvircC&dq=%22when+the+exponent+is+1%22&pg=PA32">page 31</a></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Jerome E. Kaufmann, Karen L. Schwitters, <i>Algebra for College Students</i>, Publisher Cengage Learning, 2010, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0538733543" title="Special:BookSources/0538733543">0538733543</a>, 9780538733540, 803 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-AHtC0IYMhYC&q=exponents+&pg=PA222">page 222</a></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Ramesh Bangia, <i>Dictionary of Information Technology</i>, Publisher Laxmi Publications, Ltd., 2010, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9380298153" title="Special:BookSources/9380298153">9380298153</a>, 9789380298153, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zQa5I2sHPKEC&q=exponentiation+caret&pg=PA212">page 212</a></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">George Grätzer, <i>First Steps in LaTeX</i>, Publisher Springer, 1999, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0817641327" title="Special:BookSources/0817641327">0817641327</a>, 9780817641320, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mLdg5ZdDKToC&q=subscripts+and+superscripts+caret&pg=PA17">page 17</a></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">S. Tucker Taft, Robert A. Duff, Randall L. Brukardt, Erhard Ploedereder, Pascal Leroy, <i>Ada 2005 Reference Manual</i>, Volume 4348 of Lecture Notes in Computer Science, Publisher Springer, 2007, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3540693351" title="Special:BookSources/3540693351">3540693351</a>, 9783540693352, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=694P3YtXh-0C&q=double+star+exponentiate&pg=PA12">page 13</a></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">C. Xavier, <i>Fortran 77 And Numerical Methods</i>, Publisher New Age International, 1994, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/812240670X" title="Special:BookSources/812240670X">812240670X</a>, 9788122406702, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WYMgF9WFty0C&dq=fortran+asterisk+exponentiation&pg=PA20">page 20</a></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">Randal Schwartz, Brian Foy, Tom Phoenix, <i>Learning Perl</i>, Publisher O'Reilly Media, Inc., 2011, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1449313140" title="Special:BookSources/1449313140">1449313140</a>, 9781449313142, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=l2IwEuRjeNwC&q=double+asterisk+exponentiation&pg=PA24">page 24</a></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">Matthew A. Telles, <i>Python Power!: The Comprehensive Guide</i>, Publisher Course Technology PTR, 2008, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1598631586" title="Special:BookSources/1598631586">1598631586</a>, 9781598631586, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=754knV_fyf8C&q=double+asterisk+exponentiation&pg=PA46">page 46</a></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">Kevin C. Baird, <i>Ruby by Example: Concepts and Code</i>, Publisher No Starch Press, 2007, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1593271484" title="Special:BookSources/1593271484">1593271484</a>, 9781593271480, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kq2dBNdAl3IC&q=double+asterisk+exponentiation&pg=PA72">page 72</a></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">William P. Berlinghoff, Fernando Q. Gouvêa, <i>Math through the Ages: A Gentle History for Teachers and Others</i>, Publisher MAA, 2004, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0883857367" title="Special:BookSources/0883857367">0883857367</a>, 9780883857366, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JAXNVaPt7uQC&dq=calculator+asterisk+multiplication&pg=PA75">page 75</a></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text">Thomas Sonnabend, <i>Mathematics for Teachers: An Interactive Approach for Grades K-8</i>, Publisher: Cengage Learning, 2009, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0495561665" title="Special:BookSources/0495561665">0495561665</a>, 9780495561668, 759 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gBa2GzyXCF8C&q=extends+arithmetic&pg=PR17">page xvii</a></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">Lewis Hirsch, Arthur Goodman, <i>Understanding Elementary Algebra With Geometry: A Course for College Students</i>, Publisher: Cengage Learning, 2005, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0534999727" title="Special:BookSources/0534999727">0534999727</a>, 9780534999728, 654 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jsT7kqZubvIC&dq=%22elementary+algebra%22+variables+unknown&pg=PA48">page 48</a></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text">Ron Larson, Kimberly Nolting, <i>Elementary Algebra</i>, Publisher: Cengage Learning, 2009, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0547102275" title="Special:BookSources/0547102275">0547102275</a>, 9780547102276, 622 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=U6v78M5nYKAC&q=relationships&pg=PA210">page 210</a></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text">Charles P. McKeague, <i>Elementary Algebra</i>, Publisher: Cengage Learning, 2011, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0840064217" title="Special:BookSources/0840064217">0840064217</a>, 9780840064219, 571 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=etTbP0rItQ4C&dq=%22elementary+algebra%22+commutative&pg=PA49">page 49</a></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text">Andrew Marx, <i>Shortcut Algebra I: A Quick and Easy Way to Increase Your Algebra I Knowledge and Test Scores</i>, Publisher Kaplan Publishing, 2007, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1419552880" title="Special:BookSources/1419552880">1419552880</a>, 9781419552885, 288 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=o9GYQjZ7ZwUC&q=like+terms&pg=PA51">page 51</a></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text">Mark Clark, Cynthia Anfinson, <i>Beginning Algebra: Connecting Concepts Through Applications</i>, Publisher Cengage Learning, 2011, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0534419380" title="Special:BookSources/0534419380">0534419380</a>, 9780534419387, 793 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wCzuRMC5048C&q=equation&pg=PA134">page 134</a></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text">Alan S. Tussy, R. David Gustafson, <i>Elementary and Intermediate Algebra</i>, Publisher Cengage Learning, 2012, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1111567689" title="Special:BookSources/1111567689">1111567689</a>, 9781111567682, 1163 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xqio_Xn4t7oC&dq=algebra+Pythagoras+hypotenuse&pg=PA493">page 493</a></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text">Douglas Downing, <i>Algebra the Easy Way</i>, Publisher Barron's Educational Series, 2003, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0764119729" title="Special:BookSources/0764119729">0764119729</a>, 9780764119729, 392 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RiX-TJLiQv0C&dq=algebra+equality+++reflexive++symmetric++transitive&pg=PA20">page 20</a></span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text">Ron Larson, Robert Hostetler, <i>Intermediate Algebra</i>, Publisher Cengage Learning, 2008, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0618753524" title="Special:BookSources/0618753524">0618753524</a>, 9780618753529, 857 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=b3vqad8tbiAC&dq=algebra+inequality+properties&pg=PA96">page 96</a></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/q/1043755">"What is the following property of inequality called?"</a>. <i><a href="/wiki/Stack_Exchange" title="Stack Exchange">Stack Exchange</a></i>. November 29, 2014<span class="reference-accessdate">. Retrieved <span class="nowrap">4 May</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Stack+Exchange&rft.atitle=What+is+the+following+property+of+inequality+called%3F&rft.date=2014-11-29&rft_id=https%3A%2F%2Fmath.stackexchange.com%2Fq%2F1043755&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElementary+algebra" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text">Chris Carter, <i>Physics: Facts and Practice for A Level</i>, Publisher Oxford University Press, 2001, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/019914768X" title="Special:BookSources/019914768X">019914768X</a>, 9780199147687, 144 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ff9gxZPYafcC&q=turned+around&pg=PA50">page 50</a></span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSlavin,_Steve1989" class="citation book cs1">Slavin, Steve (1989). <a rel="nofollow" class="external text" href="https://archive.org/details/allmathyoullever00slav/page/72"><i>All the Math You'll Ever Need</i></a>. <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/allmathyoullever00slav/page/72">72</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-50636-2" title="Special:BookSources/0-471-50636-2"><bdi>0-471-50636-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=All+the+Math+You%27ll+Ever+Need&rft.pages=72&rft.pub=John+Wiley+%26+Sons&rft.date=1989&rft.isbn=0-471-50636-2&rft.au=Slavin%2C+Steve&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fallmathyoullever00slav%2Fpage%2F72&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElementary+algebra" class="Z3988"></span></span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text">Sinha, <i>The Pearson Guide to Quantitative Aptitude for CAT 2/e</i>Publisher: Pearson Education India, 2010, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/8131723666" title="Special:BookSources/8131723666">8131723666</a>, 9788131723661, 599 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eOnaFSKRSR0C&q=many+solutions&pg=PA195">page 195</a></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><a href="/wiki/Cynthia_Y._Young" title="Cynthia Y. Young">Cynthia Y. Young</a>, <i>Precalculus</i>, Publisher John Wiley & Sons, 2010, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0471756849" title="Special:BookSources/0471756849">0471756849</a>, 9780471756842, 1175 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9HRLAn326zEC&dq=linear+equation++two+variables++many+solutions&pg=PA699">page 699</a></span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text">Mary Jane Sterling, <i>Algebra II For Dummies</i>, Publisher: John Wiley & Sons, 2006, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0471775819" title="Special:BookSources/0471775819">0471775819</a>, 9780471775812, 384 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_0rTMuSpTY0C&dq=quadratic+equations&pg=PA37">page 37</a></span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text">John T. Irwin, <i>The Mystery to a Solution: Poe, Borges, and the Analytic Detective Story</i>, Publisher JHU Press, 1996, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0801854660" title="Special:BookSources/0801854660">0801854660</a>, 9780801854668, 512 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jsxTenuOQKgC&dq=quadratic+quadrus&pg=PA372">page 372</a></span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text">Sharma/khattar, <i>The Pearson Guide To Objective Mathematics For Engineering Entrance Examinations, 3/E</i>, Publisher Pearson Education India, 2010, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/8131723631" title="Special:BookSources/8131723631">8131723631</a>, 9788131723630, 1248 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2v-f9x7-FlsC&dq=quadratic%20equations%20%20ax2%20%2B%20bx%20%2B%20c%20%3D%200&pg=RA13-PA33">page 621</a></span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text">Aven Choo, <i>LMAN OL Additional Maths Revision Guide 3</i>, Publisher Pearson Education South Asia, 2007, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9810600011" title="Special:BookSources/9810600011">9810600011</a>, 9789810600013, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NsBXDMrzcJIC&dq=%22+exponential+equation+%22+aX+%3D+b&pg=RA2-PA29">page 105</a></span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text">John C. Peterson, <i>Technical Mathematics With Calculus</i>, Publisher Cengage Learning, 2003, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0766861899" title="Special:BookSources/0766861899">0766861899</a>, 9780766861893, 1613 pages, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PGuSDjHvircC&dq=%22+radical+equation%22&pg=PA525">page 525</a></span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://web.mat.bham.ac.uk/C.J.Sangwin/euler/">Euler's Elements of Algebra</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110413234352/http://web.mat.bham.ac.uk/C.J.Sangwin/euler/">Archived</a> 2011-04-13 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEulerHewlettHornerBernoulli2018" class="citation web cs1">Euler, Leonhard; Hewlett, John; Horner, Francis; Bernoulli, Jean; Lagrange, Joseph Louis (4 May 2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=X8yv0sj4_1YC&q=euler+elements">"Elements of Algebra"</a>. Longman, Orme<span class="reference-accessdate">. Retrieved <span class="nowrap">4 May</span> 2018</span> – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Elements+of+Algebra&rft.pub=Longman%2C+Orme&rft.date=2018-05-04&rft.aulast=Euler&rft.aufirst=Leonhard&rft.au=Hewlett%2C+John&rft.au=Horner%2C+Francis&rft.au=Bernoulli%2C+Jean&rft.au=Lagrange%2C+Joseph+Louis&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DX8yv0sj4_1YC%26q%3Deuler%2Belements&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElementary+algebra" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elementary_algebra&action=edit&section=27" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a 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