System of linear equations

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>System of linear equations - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"cc3739b5-8a3f-42c5-b16e-25da48e4ad8b","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"System_of_linear_equations","wgTitle":"System of linear equations","wgCurRevisionId":1273779114,"wgRevisionId":1273779114,"wgArticleId":113087,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description is different from Wikidata","Use American English from February 2025","All Wikipedia articles written in American English","Use mdy dates from February 2025","Articles lacking in-text citations from October 2015","All articles lacking in-text citations","All articles with unsourced statements","Articles with unsourced statements from March 2017", "Pages displaying short descriptions of redirect targets via Module:Annotated link","Commons category link from Wikidata","Pages that use a deprecated format of the math tags","Equations","Linear algebra","Numerical linear algebra"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"System_of_linear_equations","wgRelevantArticleId":113087,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":40000, "wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q11203","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage": "ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","ext.scribunto.logs","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.16"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Secretsharing_3-point.svg/1200px-Secretsharing_3-point.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="1200"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Secretsharing_3-point.svg/800px-Secretsharing_3-point.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="800"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Secretsharing_3-point.svg/640px-Secretsharing_3-point.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="640"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="System of linear equations - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/System_of_linear_equations"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=System_of_linear_equations&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/System_of_linear_equations"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-System_of_linear_equations rootpage-System_of_linear_equations skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" title="Main menu" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li><li id="n-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages"><span>Special pages</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=System+of+linear+equations" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=System+of+linear+equations" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=System+of+linear+equations" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=System+of+linear+equations" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Elementary_examples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Elementary_examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Elementary examples</span> </div> </a> <button aria-controls="toc-Elementary_examples-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 Elementary examples subsection</span> </button> <ul id="toc-Elementary_examples-sublist" class="vector-toc-list"> <li id="toc-Trivial_example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Trivial_example"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Trivial example</span> </div> </a> <ul id="toc-Trivial_example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simple_nontrivial_example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simple_nontrivial_example"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Simple nontrivial example</span> </div> </a> <ul id="toc-Simple_nontrivial_example-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-General_form" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#General_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>General form</span> </div> </a> <button aria-controls="toc-General_form-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 General form subsection</span> </button> <ul id="toc-General_form-sublist" class="vector-toc-list"> <li id="toc-Vector_equation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vector_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Vector equation</span> </div> </a> <ul id="toc-Vector_equation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matrix_equation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matrix_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Matrix equation</span> </div> </a> <ul id="toc-Matrix_equation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Solution_set" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Solution_set"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Solution set</span> </div> </a> <button aria-controls="toc-Solution_set-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 Solution set subsection</span> </button> <ul id="toc-Solution_set-sublist" class="vector-toc-list"> <li id="toc-Geometric_interpretation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometric_interpretation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Geometric interpretation</span> </div> </a> <ul id="toc-Geometric_interpretation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_behavior" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_behavior"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>General behavior</span> </div> </a> <ul id="toc-General_behavior-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-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 Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Independence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Independence"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Independence</span> </div> </a> <ul id="toc-Independence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Consistency" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Consistency"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Consistency</span> </div> </a> <ul id="toc-Consistency-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equivalence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equivalence"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Equivalence</span> </div> </a> <ul id="toc-Equivalence-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Solving_a_linear_system" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Solving_a_linear_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Solving a linear system</span> </div> </a> <button aria-controls="toc-Solving_a_linear_system-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 a linear system subsection</span> </button> <ul id="toc-Solving_a_linear_system-sublist" class="vector-toc-list"> <li id="toc-Describing_the_solution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Describing_the_solution"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Describing the solution</span> </div> </a> <ul id="toc-Describing_the_solution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elimination_of_variables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elimination_of_variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Elimination of variables</span> </div> </a> <ul id="toc-Elimination_of_variables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Row_reduction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Row_reduction"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Row reduction</span> </div> </a> <ul id="toc-Row_reduction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cramer's_rule" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cramer's_rule"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Cramer's rule</span> </div> </a> <ul id="toc-Cramer's_rule-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matrix_solution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matrix_solution"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Matrix solution</span> </div> </a> <ul id="toc-Matrix_solution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Other methods</span> </div> </a> <ul id="toc-Other_methods-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Homogeneous_systems" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Homogeneous_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Homogeneous systems</span> </div> </a> <button aria-controls="toc-Homogeneous_systems-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 Homogeneous systems subsection</span> </button> <ul id="toc-Homogeneous_systems-sublist" class="vector-toc-list"> <li id="toc-Homogeneous_solution_set" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Homogeneous_solution_set"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Homogeneous solution set</span> </div> </a> <ul id="toc-Homogeneous_solution_set-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_nonhomogeneous_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_nonhomogeneous_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Relation to nonhomogeneous systems</span> </div> </a> <ul id="toc-Relation_to_nonhomogeneous_systems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</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"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</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" title="Table of Contents" > <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">System of linear equations</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 71 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-71" 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">71 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/Lineares_Gleichungssystem" title="Lineares Gleichungssystem – Alemannic" lang="gsw" hreflang="gsw" data-title="Lineares Gleichungssystem" 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/%D9%86%D8%B8%D8%A7%D9%85_%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A7%D8%AA_%D8%AE%D8%B7%D9%8A%D8%A9" title="نظام معادلات خطية – Arabic" lang="ar" hreflang="ar" data-title="نظام معادلات خطية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Sistema_d%27ecuacions_lineals" title="Sistema d'ecuacions lineals – Aragonese" lang="an" hreflang="an" data-title="Sistema d'ecuacions lineals" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Sistema_d%27ecuaciones_lliniales" title="Sistema d'ecuaciones lliniales – Asturian" lang="ast" hreflang="ast" data-title="Sistema d'ecuaciones lliniales" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/X%C9%99tti_t%C9%99nlikl%C9%99r_sistemi" title="Xətti tənliklər sistemi – Azerbaijani" lang="az" hreflang="az" data-title="Xətti tənliklər sistemi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D2%BA%D1%8B%D2%99%D1%8B%D2%A1%D0%BB%D1%8B_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D0%BA_%D1%82%D0%B8%D0%B3%D0%B5%D2%99%D0%BB%D3%99%D0%BC%D3%99%D0%BB%D3%99%D1%80_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0%D2%BB%D1%8B" title="Һыҙыҡлы алгебраик тигеҙләмәләр системаһы – Bashkir" lang="ba" hreflang="ba" data-title="Һыҙыҡлы алгебраик тигеҙләмәләр системаһы" 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%A1%D1%96%D1%81%D1%82%D1%8D%D0%BC%D0%B0_%D0%BB%D1%96%D0%BD%D0%B5%D0%B9%D0%BD%D1%8B%D1%85_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D1%96%D1%87%D0%BD%D1%8B%D1%85_%D1%83%D1%80%D0%B0%D1%9E%D0%BD%D0%B5%D0%BD%D0%BD%D1%8F%D1%9E" title="Сістэма лінейных алгебраічных ураўненняў – Belarusian" lang="be" hreflang="be" data-title="Сістэма лінейных алгебраічных ураўненняў" 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%A1%D1%8B%D1%81%D1%82%D1%8D%D0%BC%D0%B0_%D0%BB%D1%96%D0%BD%D0%B5%D0%B9%D0%BD%D1%8B%D1%85_%D0%B0%D0%BB%D1%8C%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D1%96%D1%87%D0%BD%D1%8B%D1%85_%D1%80%D0%B0%D1%9E%D0%BD%D0%B0%D0%BD%D1%8C%D0%BD%D1%8F%D1%9E" title="Сыстэма лінейных альгебраічных раўнаньняў – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Сыстэма лінейных альгебраічных раўнаньняў" 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%A1%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D0%BB%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%B8_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D1%8F" title="Система линейни уравнения – Bulgarian" lang="bg" hreflang="bg" data-title="Система линейни уравнения" 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/Sistem_linearnih_jedna%C4%8Dina" title="Sistem linearnih jednačina – Bosnian" lang="bs" hreflang="bs" data-title="Sistem linearnih jednačina" 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/Sistema_d%27equacions_lineals" title="Sistema d'equacions lineals – Catalan" lang="ca" hreflang="ca" data-title="Sistema d'equacions lineals" 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%9B%D0%B8%D0%BD%D0%B8%D0%BB%D0%BB%D0%B5_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%C4%83%D0%BB%D0%BB%D0%B0_%D1%82%D0%B0%D0%BD%D0%BB%C4%83%D1%85%D1%81%D0%B5%D0%BD_%D1%82%D1%8B%D1%82%C4%83%D0%BC%C4%95" title="Линилле алгебрăлла танлăхсен тытăмĕ – Chuvash" lang="cv" hreflang="cv" data-title="Линилле алгебрăлла танлăхсен тытăмĕ" 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/Soustava_line%C3%A1rn%C3%ADch_rovnic" title="Soustava lineárních rovnic – Czech" lang="cs" hreflang="cs" data-title="Soustava lineárních rovnic" 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/Lineares_Gleichungssystem" title="Lineares Gleichungssystem – German" lang="de" hreflang="de" data-title="Lineares Gleichungssystem" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Lineaarv%C3%B5rrandis%C3%BCsteem" title="Lineaarvõrrandisüsteem – Estonian" lang="et" hreflang="et" data-title="Lineaarvõrrandisüsteem" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%8D%CF%83%CF%84%CE%B7%CE%BC%CE%B1_%CE%B3%CF%81%CE%B1%CE%BC%CE%BC%CE%B9%CE%BA%CF%8E%CE%BD_%CE%B5%CE%BE%CE%B9%CF%83%CF%8E%CF%83%CE%B5%CF%89%CE%BD" title="Σύστημα γραμμικών εξισώσεων – Greek" lang="el" hreflang="el" data-title="Σύστημα γραμμικών εξισώσεων" 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/Sistema_de_ecuaciones_lineales" title="Sistema de ecuaciones lineales – Spanish" lang="es" hreflang="es" data-title="Sistema de ecuaciones lineales" 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/Sistemo_de_linearaj_ekvacioj" title="Sistemo de linearaj ekvacioj – Esperanto" lang="eo" hreflang="eo" data-title="Sistemo de linearaj ekvacioj" 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/Ekuazio_linealetako_sistema" title="Ekuazio linealetako sistema – Basque" lang="eu" hreflang="eu" data-title="Ekuazio linealetako sistema" 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%AF%D8%B3%D8%AA%DA%AF%D8%A7%D9%87_%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A7%D8%AA_%D8%AE%D8%B7%DB%8C" title="دستگاه معادلات خطی – Persian" lang="fa" hreflang="fa" data-title="دستگاه معادلات خطی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/System_of_linear_equations" title="System of linear equations – Fiji Hindi" lang="hif" hreflang="hif" data-title="System of linear equations" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Syst%C3%A8me_d%27%C3%A9quations_lin%C3%A9aires" title="Système d'équations linéaires – French" lang="fr" hreflang="fr" data-title="Système d'équations linéaires" 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/Sistema_de_ecuaci%C3%B3ns_lineais" title="Sistema de ecuacións lineais – Galician" lang="gl" hreflang="gl" data-title="Sistema de ecuacións lineais" 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%97%B0%EB%A6%BD_%EC%9D%BC%EC%B0%A8_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="연립 일차 방정식 – Korean" lang="ko" hreflang="ko" data-title="연립 일차 방정식" 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/%D4%B3%D5%AE%D5%A1%D5%B5%D5%AB%D5%B6_%D5%B0%D5%A1%D5%BE%D5%A1%D5%BD%D5%A1%D6%80%D5%B8%D6%82%D5%B4%D5%B6%D5%A5%D6%80%D5%AB_%D5%B0%D5%A1%D5%B4%D5%A1%D5%AF%D5%A1%D6%80%D5%A3" title="Գծային հավասարումների համակարգ – Armenian" lang="hy" hreflang="hy" data-title="Գծային հավասարումների համակարգ" 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%B0%E0%A5%88%E0%A4%96%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3_%E0%A4%A8%E0%A4%BF%E0%A4%95%E0%A4%BE%E0%A4%AF" title="रैखिक समीकरण निकाय – Hindi" lang="hi" hreflang="hi" data-title="रैखिक समीकरण निकाय" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Sustav_linearnih_jednad%C5%BEbi" title="Sustav linearnih jednadžbi – Croatian" lang="hr" hreflang="hr" data-title="Sustav linearnih jednadžbi" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sistem_persamaan_linear" title="Sistem persamaan linear – Indonesian" lang="id" hreflang="id" data-title="Sistem persamaan linear" 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/Systema_de_equationes_linear" title="Systema de equationes linear – Interlingua" lang="ia" hreflang="ia" data-title="Systema de equationes linear" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/L%C3%ADnulegt_j%C3%B6fnuhneppi" title="Línulegt jöfnuhneppi – Icelandic" lang="is" hreflang="is" data-title="Línulegt jöfnuhneppi" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sistema_di_equazioni_lineari" title="Sistema di equazioni lineari – Italian" lang="it" hreflang="it" data-title="Sistema di equazioni lineari" 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%9E%D7%A2%D7%A8%D7%9B%D7%AA_%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%95%D7%AA_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%95%D7%AA" title="מערכת משוואות ליניאריות – Hebrew" lang="he" hreflang="he" data-title="מערכת משוואות ליניאריות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Systema_aequationum_linearium" title="Systema aequationum linearium – Latin" lang="la" hreflang="la" data-title="Systema aequationum linearium" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Line%C4%81ru_vien%C4%81dojumu_sist%C4%93ma" title="Lineāru vienādojumu sistēma – Latvian" lang="lv" hreflang="lv" data-title="Lineāru vienādojumu sistēma" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Sistema_de_equazion_linear" title="Sistema de equazion linear – Lombard" lang="lmo" hreflang="lmo" data-title="Sistema de equazion linear" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Line%C3%A1ris_egyenletrendszer" title="Lineáris egyenletrendszer – Hungarian" lang="hu" hreflang="hu" data-title="Lineáris egyenletrendszer" 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%A1%D0%B8%D1%81%D1%82%D0%B5%D0%BC_%D0%BD%D0%B0_%D0%BB%D0%B8%D0%BD%D0%B5%D0%B0%D1%80%D0%BD%D0%B8_%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D0%BA%D0%B8" title="Систем на линеарни равенки – Macedonian" lang="mk" hreflang="mk" data-title="Систем на линеарни равенки" 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/Sistem_persamaan_linear" title="Sistem persamaan linear – Malay" lang="ms" hreflang="ms" data-title="Sistem persamaan linear" 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-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A8%D1%83%D0%B3%D0%B0%D0%BC%D0%B0%D0%BD_%D1%82%D1%8D%D0%B3%D1%88%D0%B8%D1%82%D0%B3%D1%8D%D0%BB%D0%B8%D0%B9%D0%BD_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Шугаман тэгшитгэлийн систем – Mongolian" lang="mn" hreflang="mn" data-title="Шугаман тэгшитгэлийн систем" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Stelsel_van_lineaire_vergelijkingen" title="Stelsel van lineaire vergelijkingen – Dutch" lang="nl" hreflang="nl" data-title="Stelsel van lineaire vergelijkingen" 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/%E7%B7%9A%E5%9E%8B%E6%96%B9%E7%A8%8B%E5%BC%8F%E7%B3%BB" title="線型方程式系 – Japanese" lang="ja" hreflang="ja" data-title="線型方程式系" 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/Line%C3%A6rt_ligningssystem" title="Lineært ligningssystem – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Lineært ligningssystem" 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Line%C3%A6rt_likningssystem" title="Lineært likningssystem – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Lineært likningssystem" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Sist%C3%A8ma_d%27equacions_linearas" title="Sistèma d'equacions linearas – Occitan" lang="oc" hreflang="oc" data-title="Sistèma d'equacions linearas" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B0%E0%A9%87%E0%A8%96%E0%A9%80_%E0%A8%B8%E0%A8%AE%E0%A9%80%E0%A8%95%E0%A8%B0%E0%A8%A8%E0%A8%BE%E0%A8%82_%E0%A8%A6%E0%A8%BE_%E0%A8%A4%E0%A9%B0%E0%A8%A4%E0%A8%B0" title="ਰੇਖੀ ਸਮੀਕਰਨਾਂ ਦਾ ਤੰਤਰ – Punjabi" lang="pa" hreflang="pa" data-title="ਰੇਖੀ ਸਮੀਕਰਨਾਂ ਦਾ ਤੰਤਰ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%84%DB%8C%D9%86%DB%8C%D8%B1_%D8%A7%DB%8C%DA%A9%D9%88%D8%A7%DB%8C%D8%B4%D9%86%D8%B2_%D8%AF%D8%A7_%D9%BE%D8%B1%D8%A8%D9%86%D8%AF%DA%BE" title="لینیر ایکوایشنز دا پربندھ – Western Punjabi" lang="pnb" hreflang="pnb" data-title="لینیر ایکوایشنز دا پربندھ" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Uk%C5%82ad_r%C3%B3wna%C5%84_liniowych" title="Układ równań liniowych – Polish" lang="pl" hreflang="pl" data-title="Układ równań liniowych" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Sistema_de_equa%C3%A7%C3%B5es_lineares" title="Sistema de equações lineares – Portuguese" lang="pt" hreflang="pt" data-title="Sistema de equações lineares" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Sistem_de_ecua%C8%9Bii_liniare" title="Sistem de ecuații liniare – Romanian" lang="ro" hreflang="ro" data-title="Sistem de ecuații liniare" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D0%BB%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D1%8B%D1%85_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D1%85_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B9" title="Система линейных алгебраических уравнений – Russian" lang="ru" hreflang="ru" data-title="Система линейных алгебраических уравнений" 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%92%E0%B6%9A%E0%B6%A2_%E0%B7%83%E0%B6%B8%E0%B7%93%E0%B6%9A%E0%B6%BB%E0%B6%AB_%E0%B6%B4%E0%B6%AF%E0%B7%8A%E0%B6%B0%E0%B6%AD%E0%B7%92%E0%B6%BA" title="ඒකජ සමීකරණ පද්ධතිය – Sinhala" lang="si" hreflang="si" data-title="ඒකජ සමීකරණ පද්ධතිය" 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/System_of_linear_equations" title="System of linear equations – Simple English" lang="en-simple" hreflang="en-simple" data-title="System of linear equations" 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-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D8%B3%D9%90%DA%8C%D9%90%D8%B1_%D9%85%D8%B3%D8%A7%D9%88%D8%A7%D8%AA%D9%8F%D9%86_%D8%AC%D9%88_%D8%B3%D8%B1%D8%B4%D8%AA%D9%88" title="سِڌِر مساواتُن جو سرشتو – Sindhi" lang="sd" hreflang="sd" data-title="سِڌِر مساواتُن جو سرشتو" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/S%C3%BAstava_line%C3%A1rnych_rovn%C3%ADc" title="Sústava lineárnych rovníc – Slovak" lang="sk" hreflang="sk" data-title="Sústava lineárnych rovníc" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Sistem_linearnih_ena%C4%8Db" title="Sistem linearnih enačb – Slovenian" lang="sl" hreflang="sl" data-title="Sistem linearnih enačb" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%B3%DB%8C%D8%B3%D8%AA%D9%85%DB%8C_%DA%BE%D8%A7%D9%88%DA%A9%DB%8E%D8%B4%DB%95%DB%8C_%DA%BE%DB%8E%DA%B5%DB%8C" title="سیستمی ھاوکێشەی ھێڵی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="سیستمی ھاوکێشەی ھێڵی" 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%A1%D0%B8%D1%81%D1%82%D0%B5%D0%BC_%D0%BB%D0%B8%D0%BD%D0%B5%D0%B0%D1%80%D0%BD%D0%B8%D1%85_%D1%98%D0%B5%D0%B4%D0%BD%D0%B0%D1%87%D0%B8%D0%BD%D0%B0" title="Систем линеарних једначина – Serbian" lang="sr" hreflang="sr" data-title="Систем линеарних једначина" 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-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Sistem_linearnih_jedna%C4%8Dina" title="Sistem linearnih jednačina – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Sistem linearnih jednačina" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Lineaarinen_yht%C3%A4l%C3%B6ryhm%C3%A4" title="Lineaarinen yhtälöryhmä – Finnish" lang="fi" hreflang="fi" data-title="Lineaarinen yhtälöryhmä" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Linj%C3%A4rt_ekvationssystem" title="Linjärt ekvationssystem – Swedish" lang="sv" hreflang="sv" data-title="Linjärt ekvationssystem" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AF%87%E0%AE%B0%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D_%E0%AE%9A%E0%AE%AE%E0%AE%A9%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81%E0%AE%95%E0%AE%B3%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%A4%E0%AF%8A%E0%AE%95%E0%AF%81%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81" title="நேரியல் சமன்பாடுகளின் தொகுப்பு – Tamil" lang="ta" hreflang="ta" data-title="நேரியல் சமன்பாடுகளின் தொகுப்பு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A1%D1%8B%D0%B7%D1%8B%D0%BA%D1%87%D0%B0_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D0%BA_%D1%82%D0%B8%D0%B3%D0%B5%D0%B7%D0%BB%D3%99%D0%BC%D3%99%D0%BB%D3%99%D1%80_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0%D1%81%D1%8B" title="Сызыкча алгебраик тигезләмәләр системасы – Tatar" lang="tt" hreflang="tt" data-title="Сызыкча алгебраик тигезләмәләр системасы" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B0%E0%B8%9A%E0%B8%9A%E0%B8%AA%E0%B8%A1%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B9%80%E0%B8%AA%E0%B9%89%E0%B8%99" title="ระบบสมการเชิงเส้น – Thai" lang="th" hreflang="th" data-title="ระบบสมการเชิงเส้น" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Do%C4%9Frusal_denklem_dizgesi" title="Doğrusal denklem dizgesi – Turkish" lang="tr" hreflang="tr" data-title="Doğrusal denklem dizgesi" 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%A1%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D0%BB%D1%96%D0%BD%D1%96%D0%B9%D0%BD%D0%B8%D1%85_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D1%97%D1%87%D0%BD%D0%B8%D1%85_%D1%80%D1%96%D0%B2%D0%BD%D1%8F%D0%BD%D1%8C" title="Система лінійних алгебраїчних рівнянь – Ukrainian" lang="uk" hreflang="uk" data-title="Система лінійних алгебраїчних рівнянь" 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/%DB%8C%DA%A9%D9%84%D8%AE%D8%AA_%D9%84%DA%A9%DB%8C%D8%B1%DB%8C_%D9%85%D8%B3%D8%A7%D9%88%D8%A7%D8%AA_%DA%A9%D8%A7_%D9%86%D8%B8%D8%A7%D9%85" title="یکلخت لکیری مساوات کا نظام – Urdu" lang="ur" hreflang="ur" data-title="یکلخت لکیری مساوات کا نظام" 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/H%E1%BB%87_ph%C6%B0%C6%A1ng_tr%C3%ACnh_tuy%E1%BA%BFn_t%C3%ADnh" title="Hệ phương trình tuyến tính – Vietnamese" lang="vi" hreflang="vi" data-title="Hệ phương trình tuyến tính" 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-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Sistema_han_ekawsyones_linyales" title="Sistema han ekawsyones linyales – Waray" lang="war" hreflang="war" data-title="Sistema han ekawsyones linyales" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84" title="线性方程组 – Wu" lang="wuu" hreflang="wuu" data-title="线性方程组" 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-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%B7%9A%E6%80%A7%E6%96%B9%E7%A8%8B%E7%B5%84" title="線性方程組 – Cantonese" lang="yue" hreflang="yue" data-title="線性方程組" 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/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84" title="线性方程组 – Chinese" lang="zh" hreflang="zh" data-title="线性方程组" 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/Q11203#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 aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/System_of_linear_equations" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:System_of_linear_equations" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/System_of_linear_equations"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=System_of_linear_equations&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=System_of_linear_equations&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/System_of_linear_equations"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=System_of_linear_equations&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=System_of_linear_equations&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/System_of_linear_equations" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/System_of_linear_equations" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=System_of_linear_equations&oldid=1273779114" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=System_of_linear_equations&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=System_of_linear_equations&id=1273779114&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FSystem_of_linear_equations"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FSystem_of_linear_equations"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=System_of_linear_equations&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=System_of_linear_equations&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:System_of_linear_equations" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11203" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Several equations of degree 1 to be solved simultaneously</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">October 2015</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Secretsharing_3-point.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Secretsharing_3-point.svg/220px-Secretsharing_3-point.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Secretsharing_3-point.svg/330px-Secretsharing_3-point.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Secretsharing_3-point.svg/440px-Secretsharing_3-point.svg.png 2x" data-file-width="480" data-file-height="480" /></a><figcaption>A linear system in three variables determines a collection of <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">planes</a>. The intersection point is the solution.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>system of linear equations</b> (or <b>linear system</b>) is a collection of two or more <a href="/wiki/Linear_equation" title="Linear equation">linear equations</a> involving the same <a href="/wiki/Variable_(math)" class="mw-redirect" title="Variable (math)">variables</a>.<sup id="cite_ref-FOOTNOTEAnton19872BurdenFaires1993324GolubVan_Loan199687Harper197657_1-0" class="reference"><a href="#cite_note-FOOTNOTEAnton19872BurdenFaires1993324GolubVan_Loan199687Harper197657-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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> 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}3x+2y-z=1\\2x-2y+4z=-2\\-x+{\frac {1}{2}}y-z=0\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>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo>+</mo> <mn>4</mn> <mi>z</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}3x+2y-z=1\\2x-2y+4z=-2\\-x+{\frac {1}{2}}y-z=0\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12e6e6e9dcf7b5ef5a7ddb6451332c7a248457a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:21.628ex; height:9.176ex;" alt="{\displaystyle {\begin{cases}3x+2y-z=1\\2x-2y+4z=-2\\-x+{\frac {1}{2}}y-z=0\end{cases}}}"></span></dd></dl> <p>is a system of three equations in the three variables <span class="texhtml"><i>x</i>, <i>y</i>, <i>z</i></span>. A <i><a href="/wiki/Solution_(mathematics)" class="mw-redirect" title="Solution (mathematics)">solution</a></i> to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. In the example above, a solution is given by the <a href="/wiki/Tuple" title="Tuple">ordered triple</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 (x,y,z)=(1,-2,-2),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)=(1,-2,-2),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f186a34f432b0e63ec690152e8c106290c9d9b6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.177ex; height:2.843ex;" alt="{\displaystyle (x,y,z)=(1,-2,-2),}"></span> since it makes all three equations valid. </p><p>Linear systems are a fundamental part of <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, a subject used in most modern mathematics. Computational <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> for finding the solutions are an important part of <a href="/wiki/Numerical_linear_algebra" title="Numerical linear algebra">numerical linear algebra</a>, and play a prominent role in <a href="/wiki/Engineering" title="Engineering">engineering</a>, <a href="/wiki/Physics" title="Physics">physics</a>, <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>, <a href="/wiki/Computer_science" title="Computer science">computer science</a>, and <a href="/wiki/Economics" title="Economics">economics</a>. A <a href="/wiki/Nonlinear_system" title="Nonlinear system">system of non-linear equations</a> can often be <a href="/wiki/Approximation" title="Approximation">approximated</a> by a linear system (see <a href="/wiki/Linearization" title="Linearization">linearization</a>), a helpful technique when making a <a href="/wiki/Mathematical_model" title="Mathematical model">mathematical model</a> or <a href="/wiki/Computer_simulation" title="Computer simulation">computer simulation</a> of a relatively <a href="/wiki/Complex_system" title="Complex system">complex system</a>. </p><p>Very often, and in this article, the <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> and solutions of the equations are constrained to be <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, but the theory and algorithms apply to coefficients and solutions in any <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>. For other <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structures</a>, other theories have been developed. For coefficients and solutions in an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a>, such as the <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> of <a href="/wiki/Integer" title="Integer">integers</a>, see <a href="/wiki/Linear_equation_over_a_ring" title="Linear equation over a ring">Linear equation over a ring</a>. For coefficients and solutions that are polynomials, see <a href="/wiki/Gr%C3%B6bner_basis" title="Gröbner basis">Gröbner basis</a>. For finding the "best" integer solutions among many, see <a href="/wiki/Integer_linear_programming" class="mw-redirect" title="Integer linear programming">Integer linear programming</a>. For an example of a more exotic structure to which linear algebra can be applied, see <a href="/wiki/Tropical_geometry" title="Tropical geometry">Tropical geometry</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Elementary_examples">Elementary examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=1" title="Edit section: Elementary examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Trivial_example">Trivial example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=2" title="Edit section: Trivial example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The system of one equation in one unknown </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=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fd1d3360b1e037f634df50bf0f1405acdf94c44" 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=4}"></span></dd></dl> <p>has the 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=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>However, most interesting linear systems have at least two equations. </p> <div class="mw-heading mw-heading3"><h3 id="Simple_nontrivial_example">Simple nontrivial example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=3" title="Edit section: Simple nontrivial example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The simplest kind of nontrivial linear system involves two equations and two variables: </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{alignedat}{5}2x&&\;+\;&&3y&&\;=\;&&6&\\4x&&\;+\;&&9y&&\;=\;&&15&.\end{alignedat}}}"> <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" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mn>2</mn> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>3</mn> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>6</mn> </mtd> <mtd /> </mtr> <mtr> <mtd> <mn>4</mn> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>9</mn> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>15</mn> </mtd> <mtd> <mi></mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{5}2x&&\;+\;&&3y&&\;=\;&&6&\\4x&&\;+\;&&9y&&\;=\;&&15&.\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31362c58c90d896569ebe7aee86ab835edd1fd87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.73ex; height:5.843ex;" alt="{\displaystyle {\begin{alignedat}{5}2x&&\;+\;&&3y&&\;=\;&&6&\\4x&&\;+\;&&9y&&\;=\;&&15&.\end{alignedat}}}"></span></dd></dl> <p>One method for solving such a system is as follows. First, solve the top equation 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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> in terms 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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>: </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-{\frac {3}{2}}y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>3</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=3-{\frac {3}{2}}y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5b071ac862c00ea9e6e705b862dc6da685964c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.232ex; height:5.176ex;" alt="{\displaystyle x=3-{\frac {3}{2}}y.}"></span></dd></dl> <p>Now <a href="/wiki/Substitution_(algebra)" class="mw-redirect" title="Substitution (algebra)">substitute</a> this expression for <i>x</i> into the bottom 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 4\left(3-{\frac {3}{2}}y\right)+9y=15.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>9</mn> <mi>y</mi> <mo>=</mo> <mn>15.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\left(3-{\frac {3}{2}}y\right)+9y=15.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3bb7a73e135625032208ef76d118c8c4e9aad4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.356ex; height:6.176ex;" alt="{\displaystyle 4\left(3-{\frac {3}{2}}y\right)+9y=15.}"></span></dd></dl> <p>This results in a single equation involving only the variable <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>. Solving gives <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=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f53b404b1fdd041a589f1f2425e45a2edba110" 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=1}"></span>, and substituting this back into the equation 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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> yields <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"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </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/486a5db1c61dfd01f6e5822c0ea77a97a56d2368" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.427ex; height:5.176ex;" alt="{\displaystyle x={\frac {3}{2}}}"></span>. This method generalizes to systems with additional variables (see "elimination of variables" below, or the article on <a href="/wiki/Elementary_algebra" title="Elementary algebra">elementary algebra</a>.) </p> <div class="mw-heading mw-heading2"><h2 id="General_form">General form</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=4" title="Edit section: General form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A general system of <i>m</i> linear equations with <i>n</i> <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">unknowns</a> and <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> can be written 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 {\begin{cases}a_{11}x_{1}+a_{12}x_{2}+\dots +a_{1n}x_{n}=b_{1}\\a_{21}x_{1}+a_{22}x_{2}+\dots +a_{2n}x_{n}=b_{2}\\\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\dots +a_{mn}x_{n}=b_{m},\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> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>,</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}a_{11}x_{1}+a_{12}x_{2}+\dots +a_{1n}x_{n}=b_{1}\\a_{21}x_{1}+a_{22}x_{2}+\dots +a_{2n}x_{n}=b_{2}\\\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\dots +a_{mn}x_{n}=b_{m},\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/117a60f6f40e5f9eb71ff341c22917563633a58e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:39.14ex; height:12.509ex;" alt="{\displaystyle {\begin{cases}a_{11}x_{1}+a_{12}x_{2}+\dots +a_{1n}x_{n}=b_{1}\\a_{21}x_{1}+a_{22}x_{2}+\dots +a_{2n}x_{n}=b_{2}\\\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\dots +a_{mn}x_{n}=b_{m},\end{cases}}}"></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 x_{1},x_{2},\dots ,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},x_{2},\dots ,x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75c2d357bc1b965979bf171b5ba3bac0f68961c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.528ex; height:2.009ex;" alt="{\displaystyle x_{1},x_{2},\dots ,x_{n}}"></span> are the unknowns, <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_{11},a_{12},\dots ,a_{mn}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{11},a_{12},\dots ,a_{mn}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bac3ab39e0a2b1d9d1f47af5af9e08701d3cb55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.315ex; height:2.009ex;" alt="{\displaystyle a_{11},a_{12},\dots ,a_{mn}}"></span> are the coefficients of the system, 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_{1},b_{2},\dots ,b_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{1},b_{2},\dots ,b_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65445659bf1ac1f6a0b94857747e5256cbc4b9b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.988ex; height:2.509ex;" alt="{\displaystyle b_{1},b_{2},\dots ,b_{m}}"></span> are the constant terms.<sup id="cite_ref-FOOTNOTEBeauregardFraleigh197365_3-0" class="reference"><a href="#cite_note-FOOTNOTEBeauregardFraleigh197365-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Often the coefficients and unknowns are <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, but <a href="/wiki/Integer" title="Integer">integers</a> and <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> are also seen, as are polynomials and elements of an abstract <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structure</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Vector_equation">Vector equation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=5" title="Edit section: Vector equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One extremely helpful view is that each unknown is a weight for a <a href="/wiki/Column_vector" class="mw-redirect" title="Column vector">column vector</a> in a <a href="/wiki/Linear_combination" title="Linear combination">linear combination</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_{1}{\begin{bmatrix}a_{11}\\a_{21}\\\vdots \\a_{m1}\end{bmatrix}}+x_{2}{\begin{bmatrix}a_{12}\\a_{22}\\\vdots \\a_{m2}\end{bmatrix}}+\dots +x_{n}{\begin{bmatrix}a_{1n}\\a_{2n}\\\vdots \\a_{mn}\end{bmatrix}}={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}{\begin{bmatrix}a_{11}\\a_{21}\\\vdots \\a_{m1}\end{bmatrix}}+x_{2}{\begin{bmatrix}a_{12}\\a_{22}\\\vdots \\a_{m2}\end{bmatrix}}+\dots +x_{n}{\begin{bmatrix}a_{1n}\\a_{2n}\\\vdots \\a_{mn}\end{bmatrix}}={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d31abcce8051bbd2cd60f41a61a17b35b2cf383c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:51.085ex; height:13.843ex;" alt="{\displaystyle x_{1}{\begin{bmatrix}a_{11}\\a_{21}\\\vdots \\a_{m1}\end{bmatrix}}+x_{2}{\begin{bmatrix}a_{12}\\a_{22}\\\vdots \\a_{m2}\end{bmatrix}}+\dots +x_{n}{\begin{bmatrix}a_{1n}\\a_{2n}\\\vdots \\a_{mn}\end{bmatrix}}={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}}}"></span></dd></dl> <p>This allows all the language and theory of <i><a href="/wiki/Vector_space" title="Vector space">vector spaces</a></i> (or more generally, <i><a href="/wiki/Module_(mathematics)" title="Module (mathematics)">modules</a></i>) to be brought to bear. For example, the collection of all possible linear combinations of the vectors on the <a href="/wiki/Sides_of_an_equation" title="Sides of an equation">left-hand side</a> (LHS) is called their <i><a href="/wiki/Span_(linear_algebra)" class="mw-redirect" title="Span (linear algebra)">span</a></i>, and the equations have a solution just when the right-hand vector is within that span. If every vector within that span has exactly one expression as a linear combination of the given left-hand vectors, then any solution is unique. In any event, the span has a <i><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a></i> of <a href="/wiki/Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a> vectors that do guarantee exactly one expression; and the number of vectors in that basis (its <i><a href="/wiki/Dimension_(linear_algebra)" class="mw-redirect" title="Dimension (linear algebra)">dimension</a></i>) cannot be larger than <i>m</i> or <i>n</i>, but it can be smaller. This is important because if we have <i>m</i> independent vectors a solution is guaranteed regardless of the right-hand side (RHS), and otherwise not guaranteed. </p> <div class="mw-heading mw-heading3"><h3 id="Matrix_equation">Matrix equation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=6" title="Edit section: Matrix equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The vector equation is equivalent to a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> equation of the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mathbf {x} =\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {x} =\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d894430af69e29d6dda5aacbf4bb19336226a0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.738ex; height:2.176ex;" alt="{\displaystyle A\mathbf {x} =\mathbf {b} }"></span> where <i>A</i> is an <i>m</i>×<i>n</i> matrix, <b>x</b> is a <a href="/wiki/Column_vector" class="mw-redirect" title="Column vector">column vector</a> with <i>n</i> entries, and <b>b</b> is a column vector with <i>m</i> entries.<sup id="cite_ref-FOOTNOTEBeauregardFraleigh197365–66_4-0" class="reference"><a href="#cite_note-FOOTNOTEBeauregardFraleigh197365–66-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{bmatrix}},\quad \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}},\quad \mathbf {b} ={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{bmatrix}},\quad \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}},\quad \mathbf {b} ={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcdcd3fc5a04d379131e89f974de46b762eecc61" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:59.363ex; height:14.176ex;" alt="{\displaystyle A={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{bmatrix}},\quad \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}},\quad \mathbf {b} ={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}}.}"></span> The number of vectors in a basis for the span is now expressed as the <i><a href="/wiki/Rank_(linear_algebra)" title="Rank (linear algebra)">rank</a></i> of the matrix. </p> <div class="mw-heading mw-heading2"><h2 id="Solution_set">Solution set</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=7" title="Edit section: Solution set"><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: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="nowrap"><i>x</i> − <i>y</i> = −1</span> and <span class="nowrap">3<i>x</i> + <i>y</i> = 9</span> is the single point (2, 3).</figcaption></figure> <p>A <i><a href="/wiki/Solution_(mathematics)" class="mw-redirect" title="Solution (mathematics)">solution</a></i> of a linear system is an assignment of values to the variables <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_{2},\dots ,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},x_{2},\dots ,x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75c2d357bc1b965979bf171b5ba3bac0f68961c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.528ex; height:2.009ex;" alt="{\displaystyle x_{1},x_{2},\dots ,x_{n}}"></span> such that each of the equations is satisfied. The <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of all possible solutions is called the <i><a href="/wiki/Solution_set" title="Solution set">solution set</a></i>.<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> </p><p>A linear system may behave in any one of three possible ways: </p> <ol><li>The system has <i>infinitely many solutions</i>.</li> <li>The system has a <i>unique solution</i>.</li> <li>The system has <i>no solution</i>.</li></ol> <div class="mw-heading mw-heading3"><h3 id="Geometric_interpretation">Geometric interpretation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=8" title="Edit section: Geometric interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a system involving two variables (<i>x</i> and <i>y</i>), each linear equation determines a <a href="/wiki/Line_(mathematics)" class="mw-redirect" title="Line (mathematics)">line</a> on the <i>xy</i>-<a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">plane</a>. Because a solution to a linear system must satisfy all of the equations, the solution set is the <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> of these lines, and is hence either a line, a single point, or the <a href="/wiki/Empty_set" title="Empty set">empty set</a>. </p><p>For three variables, each linear equation determines a <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">plane</a> in <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a>, and the solution set is the intersection of these planes. Thus the solution set may be a plane, a line, a single point, or the empty set. For example, as three parallel planes do not have a common point, the solution set of their equations is empty; the solution set of the equations of three planes intersecting at a point is single point; if three planes pass through two points, their equations have at least two common solutions; in fact the solution set is infinite and consists in all the line passing through these points.<sup id="cite_ref-FOOTNOTECullen19903_6-0" class="reference"><a href="#cite_note-FOOTNOTECullen19903-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>For <i>n</i> variables, each linear equation determines a <a href="/wiki/Hyperplane" title="Hyperplane">hyperplane</a> in <a href="/wiki/N-dimensional_space" class="mw-redirect" title="N-dimensional space"><i>n</i>-dimensional space</a>. The solution set is the intersection of these hyperplanes, and is a <a href="/wiki/Flat_(geometry)" title="Flat (geometry)">flat</a>, which may have any dimension lower than <i>n</i>. </p> <div class="mw-heading mw-heading3"><h3 id="General_behavior">General behavior</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=9" title="Edit section: General behavior"><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:Intersecting_Planes_2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Intersecting_Planes_2.svg/220px-Intersecting_Planes_2.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Intersecting_Planes_2.svg/330px-Intersecting_Planes_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Intersecting_Planes_2.svg/440px-Intersecting_Planes_2.svg.png 2x" data-file-width="480" data-file-height="480" /></a><figcaption>The solution set for two equations in three variables is, in general, a line.</figcaption></figure> <p>In general, the behavior of a linear system is determined by the relationship between the number of equations and the number of unknowns. Here, "in general" means that a different behavior may occur for specific values of the coefficients of the equations. </p> <ul><li>In general, a system with fewer equations than unknowns has infinitely many solutions, but it may have no solution. Such a system is known as an <a href="/wiki/Underdetermined_system" title="Underdetermined system">underdetermined system</a>.</li> <li>In general, a system with the same number of equations and unknowns has a single unique solution.</li> <li>In general, a system with more equations than unknowns has no solution. Such a system is also known as an <a href="/wiki/Overdetermined_system" title="Overdetermined system">overdetermined system</a>.</li></ul> <p>In the first case, the <a href="/wiki/Dimension" title="Dimension">dimension</a> of the solution set is, in general, equal to <span class="nowrap"><i>n</i> − <i>m</i></span>, where <i>n</i> is the number of variables and <i>m</i> is the number of equations. </p><p>The following pictures illustrate this trichotomy in the case of two variables: </p> <dl><dd><table class="wikitable"> <tbody><tr> <td width="150" align="center"><span typeof="mw:File"><a href="/wiki/File:One_Line.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/One_Line.svg/120px-One_Line.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/One_Line.svg/180px-One_Line.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/One_Line.svg/240px-One_Line.svg.png 2x" data-file-width="500" data-file-height="500" /></a></span> </td> <td width="150" align="center"><span typeof="mw:File"><a href="/wiki/File:Two_Lines.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Two_Lines.svg/120px-Two_Lines.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Two_Lines.svg/180px-Two_Lines.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Two_Lines.svg/240px-Two_Lines.svg.png 2x" data-file-width="500" data-file-height="500" /></a></span> </td> <td width="150" align="center"><span typeof="mw:File"><a href="/wiki/File:Three_Lines.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Three_Lines.svg/120px-Three_Lines.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Three_Lines.svg/180px-Three_Lines.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Three_Lines.svg/240px-Three_Lines.svg.png 2x" data-file-width="546" data-file-height="546" /></a></span> </td></tr> <tr> <td align="center">One equation </td> <td align="center">Two equations </td> <td align="center">Three equations </td></tr></tbody></table></dd></dl> <p>The first system has infinitely many solutions, namely all of the points on the blue line. The second system has a single unique solution, namely the intersection of the two lines. The third system has no solutions, since the three lines share no common point. </p><p>It must be kept in mind that the pictures above show only the most common case (the general case). It is possible for a system of two equations and two unknowns to have no solution (if the two lines are parallel), or for a system of three equations and two unknowns to be solvable (if the three lines intersect at a single point). </p><p>A system of linear equations behave differently from the general case if the equations are <i><a href="/wiki/Linear_independence" title="Linear independence">linearly dependent</a></i>, or if it is <i><a href="#Consistency">inconsistent</a></i> and has no more equations than unknowns. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=10" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Independence">Independence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=11" title="Edit section: Independence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The equations of a linear system are <b>independent</b> if none of the equations can be derived algebraically from the others. When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set. For linear equations, logical independence is the same as <a href="/wiki/Linear_independence" title="Linear independence">linear independence</a>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Three_Intersecting_Lines.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Three_Intersecting_Lines.svg/220px-Three_Intersecting_Lines.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Three_Intersecting_Lines.svg/330px-Three_Intersecting_Lines.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Three_Intersecting_Lines.svg/440px-Three_Intersecting_Lines.svg.png 2x" data-file-width="1089" data-file-height="1089" /></a><figcaption>The equations <span class="nowrap"><i>x</i> − 2<i>y</i> = −1</span>, <span class="nowrap">3<i>x</i> + 5<i>y</i> = 8</span>, and <span class="nowrap">4<i>x</i> + 3<i>y</i> = 7</span> are linearly dependent.</figcaption></figure> <p>For example, the equations </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 3x+2y=6\;\;\;\;{\text{and}}\;\;\;\;6x+4y=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>6</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mn>6</mn> <mi>x</mi> <mo>+</mo> <mn>4</mn> <mi>y</mi> <mo>=</mo> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x+2y=6\;\;\;\;{\text{and}}\;\;\;\;6x+4y=12}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34514061fa890d7aeb05db114013d4631de46df1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.894ex; height:2.509ex;" alt="{\displaystyle 3x+2y=6\;\;\;\;{\text{and}}\;\;\;\;6x+4y=12}"></span></dd></dl> <p>are not independent — they are the same equation when scaled by a factor of two, and they would produce identical graphs. This is an example of equivalence in a system of linear equations. </p><p>For a more complicated example, the equations </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{alignedat}{5}x&&\;-\;&&2y&&\;=\;&&-1&\\3x&&\;+\;&&5y&&\;=\;&&8&\\4x&&\;+\;&&3y&&\;=\;&&7&\end{alignedat}}}"> <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" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>−</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>2</mn> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd /> </mtr> <mtr> <mtd> <mn>3</mn> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>5</mn> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>8</mn> </mtd> <mtd /> </mtr> <mtr> <mtd> <mn>4</mn> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>3</mn> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>7</mn> </mtd> <mtd /> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{5}x&&\;-\;&&2y&&\;=\;&&-1&\\3x&&\;+\;&&5y&&\;=\;&&8&\\4x&&\;+\;&&3y&&\;=\;&&7&\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d26b40c273dff9c610f2f279460dff990412e49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:14.729ex; height:8.843ex;" alt="{\displaystyle {\begin{alignedat}{5}x&&\;-\;&&2y&&\;=\;&&-1&\\3x&&\;+\;&&5y&&\;=\;&&8&\\4x&&\;+\;&&3y&&\;=\;&&7&\end{alignedat}}}"></span></dd></dl> <p>are not independent, because the third equation is the sum of the other two. Indeed, any one of these equations can be derived from the other two, and any one of the equations can be removed without affecting the solution set. The graphs of these equations are three lines that intersect at a single point. </p> <div class="mw-heading mw-heading3"><h3 id="Consistency">Consistency</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=12" title="Edit section: Consistency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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">See also: <a href="/wiki/Consistent_and_inconsistent_equations" title="Consistent and inconsistent equations">Consistent and inconsistent equations</a></div> <figure class="mw-default-size" 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="nowrap">3<i>x</i> + 2<i>y</i> = 6</span> and <span class="nowrap">3<i>x</i> + 2<i>y</i> = 12</span> are inconsistent.</figcaption></figure> <p>A linear system is <b>inconsistent</b> if it has no solution, and otherwise, it is said to be <b>consistent</b>.<sup id="cite_ref-FOOTNOTEWhitelaw1991[httpsbooksgooglecombooksid6M_kDzA7-qICpgPA70_70]_7-0" class="reference"><a href="#cite_note-FOOTNOTEWhitelaw1991[httpsbooksgooglecombooksid6M_kDzA7-qICpgPA70_70]-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> When the system is inconsistent, it is possible to derive a <a href="/wiki/Contradiction" title="Contradiction">contradiction</a> from the equations, that may always be rewritten as the statement <span class="nowrap">0 = 1</span>. </p><p>For example, the equations </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 3x+2y=6\;\;\;\;{\text{and}}\;\;\;\;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>6</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <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=6\;\;\;\;{\text{and}}\;\;\;\;3x+2y=12}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/806109fad15e1c5fa55d1ac652fa7f833652df11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.894ex; height:2.509ex;" alt="{\displaystyle 3x+2y=6\;\;\;\;{\text{and}}\;\;\;\;3x+2y=12}"></span></dd></dl> <p>are inconsistent. In fact, by subtracting the first equation from the second one and multiplying both sides of the result by 1/6, we get <span class="nowrap">0 = 1</span>. The graphs of these equations on the <i>xy</i>-plane are a pair of <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallel</a> lines. </p><p>It is possible for three linear equations to be inconsistent, even though any two of them are consistent together. For example, the equations </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{alignedat}{7}x&&\;+\;&&y&&\;=\;&&1&\\2x&&\;+\;&&y&&\;=\;&&1&\\3x&&\;+\;&&2y&&\;=\;&&3&\end{alignedat}}}"> <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 right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>1</mn> </mtd> <mtd /> </mtr> <mtr> <mtd> <mn>2</mn> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>1</mn> </mtd> <mtd /> </mtr> <mtr> <mtd> <mn>3</mn> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>2</mn> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>3</mn> </mtd> <mtd /> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{7}x&&\;+\;&&y&&\;=\;&&1&\\2x&&\;+\;&&y&&\;=\;&&1&\\3x&&\;+\;&&2y&&\;=\;&&3&\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9387db0d00da512472d4814bb75481d44c09413e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:12.921ex; height:8.843ex;" alt="{\displaystyle {\begin{alignedat}{7}x&&\;+\;&&y&&\;=\;&&1&\\2x&&\;+\;&&y&&\;=\;&&1&\\3x&&\;+\;&&2y&&\;=\;&&3&\end{alignedat}}}"></span></dd></dl> <p>are inconsistent. Adding the first two equations together gives <span class="nowrap">3<i>x</i> + 2<i>y</i> = 2</span>, which can be subtracted from the third equation to yield <span class="nowrap">0 = 1</span>. Any two of these equations have a common solution. The same phenomenon can occur for any number of equations. </p><p>In general, inconsistencies occur if the left-hand sides of the equations in a system are linearly dependent, and the constant terms do not satisfy the dependence relation. A system of equations whose left-hand sides are linearly independent is always consistent. </p><p>Putting it another way, according to the <a href="/wiki/Rouch%C3%A9%E2%80%93Capelli_theorem" title="Rouché–Capelli theorem">Rouché–Capelli theorem</a>, any system of equations (overdetermined or otherwise) is inconsistent if the <a href="/wiki/Rank_(linear_algebra)" title="Rank (linear algebra)">rank</a> of the <a href="/wiki/Augmented_matrix" title="Augmented matrix">augmented matrix</a> is greater than the rank of the <a href="/wiki/Coefficient_matrix" title="Coefficient matrix">coefficient matrix</a>. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has <i>k</i> free parameters where <i>k</i> is the difference between the number of variables and the rank; hence in such a case there is an infinitude of solutions. The rank of a system of equations (that is, the rank of the augmented matrix) can never be higher than [the number of variables] + 1, which means that a system with any number of equations can always be reduced to a system that has a number of <a href="/wiki/Independent_equation" title="Independent equation">independent equations</a> that is at most equal to [the number of variables] + 1. </p> <div class="mw-heading mw-heading3"><h3 id="Equivalence">Equivalence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=13" title="Edit section: Equivalence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Two linear systems using the same set of variables are <b>equivalent</b> if each of the equations in the second system can be derived algebraically from the equations in the first system, and vice versa. Two systems are equivalent if either both are inconsistent or each equation of each of them is a linear combination of the equations of the other one. It follows that two linear systems are equivalent if and only if they have the same solution set. </p> <div class="mw-heading mw-heading2"><h2 id="Solving_a_linear_system">Solving a linear system</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=14" title="Edit section: Solving a linear system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are several <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> for <a href="/wiki/Equation_solving" title="Equation solving">solving</a> a system of linear equations. </p> <div class="mw-heading mw-heading3"><h3 id="Describing_the_solution">Describing the solution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=15" title="Edit section: Describing the solution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When the solution set is finite, it is reduced to a single element. In this case, the unique solution is described by a sequence of equations whose left-hand sides are the names of the unknowns and right-hand sides are the corresponding values, 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=3,\;y=-2,\;z=6)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mi>z</mi> <mo>=</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x=3,\;y=-2,\;z=6)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25a9f11f417cff000b5540ed42ea1df8da77a628" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.332ex; height:2.843ex;" alt="{\displaystyle (x=3,\;y=-2,\;z=6)}"></span>. When an order on the unknowns has been fixed, for example the <a href="/wiki/Alphabetical_order" title="Alphabetical order">alphabetical order</a> the solution may be described as a <a href="/wiki/Vector_space" title="Vector space">vector</a> of values, like <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,\,6)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>6</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3,\,-2,\,6)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd044b71a38662829ab5e27ea15f154219466aa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.947ex; height:2.843ex;" alt="{\displaystyle (3,\,-2,\,6)}"></span> for the previous example. </p><p>To describe a set with an infinite number of solutions, typically some of the variables are designated as <b>free</b> (or <b>independent</b>, or as <b>parameters</b>), meaning that they are allowed to take any value, while the remaining variables are <b>dependent</b> on the values of the free variables. </p><p>For example, consider the following 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{alignedat}{7}x&&\;+\;&&3y&&\;-\;&&2z&&\;=\;&&5&\\3x&&\;+\;&&5y&&\;+\;&&6z&&\;=\;&&7&\end{alignedat}}}"> <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 right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>3</mn> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>−</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>2</mn> <mi>z</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>5</mn> </mtd> <mtd /> </mtr> <mtr> <mtd> <mn>3</mn> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>5</mn> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>6</mn> <mi>z</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>7</mn> </mtd> <mtd /> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{7}x&&\;+\;&&3y&&\;-\;&&2z&&\;=\;&&5&\\3x&&\;+\;&&5y&&\;+\;&&6z&&\;=\;&&7&\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2df5186f258524f2c10228557ba350aca8b473b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.27ex; height:5.843ex;" alt="{\displaystyle {\begin{alignedat}{7}x&&\;+\;&&3y&&\;-\;&&2z&&\;=\;&&5&\\3x&&\;+\;&&5y&&\;+\;&&6z&&\;=\;&&7&\end{alignedat}}}"></span></dd></dl> <p>The solution set to this system can be described by the following equations: </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=-7z-1\;\;\;\;{\text{and}}\;\;\;\;y=3z+2{\text{.}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>7</mn> <mi>z</mi> <mo>−</mo> <mn>1</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>y</mi> <mo>=</mo> <mn>3</mn> <mi>z</mi> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>.</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=-7z-1\;\;\;\;{\text{and}}\;\;\;\;y=3z+2{\text{.}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02f87078e85c17cfa1bfa549c6f30b0b8464b462" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.553ex; height:2.509ex;" alt="{\displaystyle x=-7z-1\;\;\;\;{\text{and}}\;\;\;\;y=3z+2{\text{.}}}"></span></dd></dl> <p>Here <i>z</i> is the free variable, while <i>x</i> and <i>y</i> are dependent on <i>z</i>. Any point in the solution set can be obtained by first choosing a value for <i>z</i>, and then computing the corresponding values for <i>x</i> and <i>y</i>. </p><p>Each free variable gives the solution space one <a href="/wiki/Degree_of_freedom" class="mw-redirect" title="Degree of freedom">degree of freedom</a>, the number of which is equal to the <a href="/wiki/Dimension" title="Dimension">dimension</a> of the solution set. For example, the solution set for the above equation is a line, since a point in the solution set can be chosen by specifying the value of the parameter <i>z</i>. An infinite solution of higher order may describe a plane, or higher-dimensional set. </p><p>Different choices for the free variables may lead to different descriptions of the same solution set. For example, the solution to the above equations can alternatively be described as follows: </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=-{\frac {3}{7}}x+{\frac {11}{7}}\;\;\;\;{\text{and}}\;\;\;\;z=-{\frac {1}{7}}x-{\frac {1}{7}}{\text{.}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>7</mn> </mfrac> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>11</mn> <mn>7</mn> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>z</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>.</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=-{\frac {3}{7}}x+{\frac {11}{7}}\;\;\;\;{\text{and}}\;\;\;\;z=-{\frac {1}{7}}x-{\frac {1}{7}}{\text{.}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4744e8c8a4cd5a82f44c4bc9bdf1797cf5fcde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:39.109ex; height:5.343ex;" alt="{\displaystyle y=-{\frac {3}{7}}x+{\frac {11}{7}}\;\;\;\;{\text{and}}\;\;\;\;z=-{\frac {1}{7}}x-{\frac {1}{7}}{\text{.}}}"></span></dd></dl> <p>Here <i>x</i> is the free variable, and <i>y</i> and <i>z</i> are dependent. </p> <div class="mw-heading mw-heading3"><h3 id="Elimination_of_variables">Elimination of variables</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=16" title="Edit section: Elimination of variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The simplest method for solving a system of linear equations is to repeatedly eliminate variables. This method can be described as follows: </p> <ol><li>In the first equation, solve for one of the variables in terms of the others.</li> <li>Substitute this expression into the remaining equations. This yields a system of equations with one fewer equation and unknown.</li> <li>Repeat steps 1 and 2 until the system is reduced to a single linear equation.</li> <li>Solve this equation, and then back-substitute until the entire solution is found.</li></ol> <p>For example, consider the following 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}x+3y-2z=5\\3x+5y+6z=7\\2x+4y+3z=8\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>3</mn> <mi>y</mi> <mo>−</mo> <mn>2</mn> <mi>z</mi> <mo>=</mo> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>5</mn> <mi>y</mi> <mo>+</mo> <mn>6</mn> <mi>z</mi> <mo>=</mo> <mn>7</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>4</mn> <mi>y</mi> <mo>+</mo> <mn>3</mn> <mi>z</mi> <mo>=</mo> <mn>8</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+3y-2z=5\\3x+5y+6z=7\\2x+4y+3z=8\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae643afaed8d321baea42c52dddc7fe2b511667a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:19.82ex; height:8.509ex;" alt="{\displaystyle {\begin{cases}x+3y-2z=5\\3x+5y+6z=7\\2x+4y+3z=8\end{cases}}}"></span></dd></dl> <p>Solving the first equation for <i>x</i> gives <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+2z-3y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>5</mn> <mo>+</mo> <mn>2</mn> <mi>z</mi> <mo>−</mo> <mn>3</mn> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=5+2z-3y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af24710c8b7fd02c9ec11ad289e5767d6f2bba64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.84ex; height:2.509ex;" alt="{\displaystyle x=5+2z-3y}"></span>, and plugging this into the second and third equation yields </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=3z+2\\y={\tfrac {7}{2}}z+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> <mi>y</mi> <mo>=</mo> <mn>3</mn> <mi>z</mi> <mo>+</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>7</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>z</mi> <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}y=3z+2\\y={\tfrac {7}{2}}z+1\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e569e09e816d25c905cdcb7ced94c40fd7681ffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.498ex; height:6.509ex;" alt="{\displaystyle {\begin{cases}y=3z+2\\y={\tfrac {7}{2}}z+1\end{cases}}}"></span></dd></dl> <p>Since the LHS of both of these equations equal <i>y</i>, equating the RHS of the equations. We now have: </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}3z+2={\tfrac {7}{2}}z+1\\\Rightarrow z=2\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>3</mn> <mi>z</mi> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>7</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>z</mi> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">⇒</mo> <mi>z</mi> <mo>=</mo> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}3z+2={\tfrac {7}{2}}z+1\\\Rightarrow z=2\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8eef2d42dfc6ebaf53c0a81e4431dbed4504ccb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:16.852ex; height:6.676ex;" alt="{\displaystyle {\begin{aligned}3z+2={\tfrac {7}{2}}z+1\\\Rightarrow z=2\end{aligned}}}"></span></dd></dl> <p>Substituting <i>z</i> = 2 into the second or third equation gives <i>y</i> = 8, and the values of <i>y</i> and <i>z</i> into the first equation yields <i>x</i> = −15. Therefore, the solution set is the ordered triple <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,z)=(-15,8,2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>15</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)=(-15,8,2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea0a8632950c312433528cd6bc4c7c2d273b2d98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.884ex; height:2.843ex;" alt="{\displaystyle (x,y,z)=(-15,8,2)}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Row_reduction">Row reduction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=17" title="Edit section: Row reduction"><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/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a></div> <p>In <b>row reduction</b> (also known as <b>Gaussian elimination</b>), the linear system is represented as an <a href="/wiki/Augmented_matrix" title="Augmented matrix">augmented matrix</a><sup id="cite_ref-FOOTNOTEBeauregardFraleigh197368_8-0" class="reference"><a href="#cite_note-FOOTNOTEBeauregardFraleigh197368-8"><span class="cite-bracket">[</span>8<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 \left[{\begin{array}{rrr|r}1&3&-2&5\\3&5&6&7\\2&4&3&8\end{array}}\right]{\text{.}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right right right" rowspacing="4pt" columnspacing="1em" columnlines="none none solid"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>7</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>.</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{array}{rrr|r}1&3&-2&5\\3&5&6&7\\2&4&3&8\end{array}}\right]{\text{.}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d99c79eb45b325d779be9693c613d9aec07b6d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:20.17ex; height:10.176ex;" alt="{\displaystyle \left[{\begin{array}{rrr|r}1&3&-2&5\\3&5&6&7\\2&4&3&8\end{array}}\right]{\text{.}}}"></span></dd></dl> <p>This matrix is then modified using <a href="/wiki/Elementary_row_operations" class="mw-redirect" title="Elementary row operations">elementary row operations</a> until it reaches <a href="/wiki/Reduced_row_echelon_form" class="mw-redirect" title="Reduced row echelon form">reduced row echelon form</a>. There are three types of elementary row operations:<sup id="cite_ref-FOOTNOTEBeauregardFraleigh197368_8-1" class="reference"><a href="#cite_note-FOOTNOTEBeauregardFraleigh197368-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><b>Type 1</b>: Swap the positions of two rows.</dd> <dd><b>Type 2</b>: Multiply a row by a nonzero <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a>.</dd> <dd><b>Type 3</b>: Add to one row a scalar multiple of another.</dd></dl> <p>Because these operations are reversible, the augmented matrix produced always represents a linear system that is equivalent to the original. </p><p>There are several specific algorithms to row-reduce an augmented matrix, the simplest of which are <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a> and <a href="/wiki/Gauss%E2%80%93Jordan_elimination" class="mw-redirect" title="Gauss–Jordan elimination">Gauss–Jordan elimination</a>. The following computation shows Gauss–Jordan elimination applied to the matrix above: </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}\left[{\begin{array}{rrr|r}1&3&-2&5\\3&5&6&7\\2&4&3&8\end{array}}\right]&\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&-4&12&-8\\2&4&3&8\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&-4&12&-8\\0&-2&7&-2\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&1&-3&2\\0&-2&7&-2\end{array}}\right]\\&\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&1&-3&2\\0&0&1&2\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&1&0&8\\0&0&1&2\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&3&0&9\\0&1&0&8\\0&0&1&2\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&0&0&-15\\0&1&0&8\\0&0&1&2\end{array}}\right].\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> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right right right" rowspacing="4pt" columnspacing="1em" columnlines="none none solid"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>7</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>∼</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right right right" rowspacing="4pt" columnspacing="1em" columnlines="none none solid"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>4</mn> </mtd> <mtd> <mn>12</mn> </mtd> <mtd> <mo>−</mo> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>∼</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right right right" rowspacing="4pt" columnspacing="1em" columnlines="none none solid"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>4</mn> </mtd> <mtd> <mn>12</mn> </mtd> <mtd> <mo>−</mo> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>∼</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right right right" rowspacing="4pt" columnspacing="1em" columnlines="none none solid"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>∼</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right right right" rowspacing="4pt" columnspacing="1em" columnlines="none none solid"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>∼</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right right right" rowspacing="4pt" columnspacing="1em" columnlines="none none solid"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>∼</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right right right" rowspacing="4pt" columnspacing="1em" columnlines="none none solid"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>9</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>∼</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right right right" rowspacing="4pt" columnspacing="1em" columnlines="none none solid"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>15</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left[{\begin{array}{rrr|r}1&3&-2&5\\3&5&6&7\\2&4&3&8\end{array}}\right]&\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&-4&12&-8\\2&4&3&8\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&-4&12&-8\\0&-2&7&-2\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&1&-3&2\\0&-2&7&-2\end{array}}\right]\\&\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&1&-3&2\\0&0&1&2\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&1&0&8\\0&0&1&2\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&3&0&9\\0&1&0&8\\0&0&1&2\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&0&0&-15\\0&1&0&8\\0&0&1&2\end{array}}\right].\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f6367f306a7947555dd25f9b3b29a5903efdabb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.838ex; width:109.213ex; height:20.843ex;" alt="{\displaystyle {\begin{aligned}\left[{\begin{array}{rrr|r}1&3&-2&5\\3&5&6&7\\2&4&3&8\end{array}}\right]&\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&-4&12&-8\\2&4&3&8\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&-4&12&-8\\0&-2&7&-2\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&1&-3&2\\0&-2&7&-2\end{array}}\right]\\&\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&1&-3&2\\0&0&1&2\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&1&0&8\\0&0&1&2\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&3&0&9\\0&1&0&8\\0&0&1&2\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&0&0&-15\\0&1&0&8\\0&0&1&2\end{array}}\right].\end{aligned}}}"></span></dd></dl> <p>The last matrix is in reduced row echelon form, and represents the system <span class="nowrap"><i>x</i> = −15</span>, <span class="nowrap"><i>y</i> = 8</span>, <span class="nowrap"><i>z</i> = 2</span>. A comparison with the example in the previous section on the algebraic elimination of variables shows that these two methods are in fact the same; the difference lies in how the computations are written down. </p> <div class="mw-heading mw-heading3"><h3 id="Cramer's_rule"><span id="Cramer.27s_rule"></span>Cramer's rule</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=18" title="Edit section: Cramer's rule"><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/Cramer%27s_rule" title="Cramer's rule">Cramer's rule</a></div> <p><b>Cramer's rule</b> is an explicit formula for the solution of a system of linear equations, with each variable given by a quotient of two <a href="/wiki/Determinant" title="Determinant">determinants</a>.<sup id="cite_ref-FOOTNOTESterling2009[httpsbooksgooglecombooksidPsNJ1alC-bsCpgPA235_235]_9-0" class="reference"><a href="#cite_note-FOOTNOTESterling2009[httpsbooksgooglecombooksidPsNJ1alC-bsCpgPA235_235]-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> For example, the solution to 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{alignedat}{7}x&\;+&\;3y&\;-&\;2z&\;=&\;5\\3x&\;+&\;5y&\;+&\;6z&\;=&\;7\\2x&\;+&\;4y&\;+&\;3z&\;=&\;8\end{alignedat}}}"> <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 right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mspace width="thickmathspace" /> <mo>+</mo> </mtd> <mtd> <mspace width="thickmathspace" /> <mn>3</mn> <mi>y</mi> </mtd> <mtd> <mi></mi> <mspace width="thickmathspace" /> <mo>−</mo> </mtd> <mtd> <mspace width="thickmathspace" /> <mn>2</mn> <mi>z</mi> </mtd> <mtd> <mi></mi> <mspace width="thickmathspace" /> <mo>=</mo> </mtd> <mtd> <mspace width="thickmathspace" /> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> <mi>x</mi> </mtd> <mtd> <mi></mi> <mspace width="thickmathspace" /> <mo>+</mo> </mtd> <mtd> <mspace width="thickmathspace" /> <mn>5</mn> <mi>y</mi> </mtd> <mtd> <mi></mi> <mspace width="thickmathspace" /> <mo>+</mo> </mtd> <mtd> <mspace width="thickmathspace" /> <mn>6</mn> <mi>z</mi> </mtd> <mtd> <mi></mi> <mspace width="thickmathspace" /> <mo>=</mo> </mtd> <mtd> <mspace width="thickmathspace" /> <mn>7</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>x</mi> </mtd> <mtd> <mi></mi> <mspace width="thickmathspace" /> <mo>+</mo> </mtd> <mtd> <mspace width="thickmathspace" /> <mn>4</mn> <mi>y</mi> </mtd> <mtd> <mi></mi> <mspace width="thickmathspace" /> <mo>+</mo> </mtd> <mtd> <mspace width="thickmathspace" /> <mn>3</mn> <mi>z</mi> </mtd> <mtd> <mi></mi> <mspace width="thickmathspace" /> <mo>=</mo> </mtd> <mtd> <mspace width="thickmathspace" /> <mn>8</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{7}x&\;+&\;3y&\;-&\;2z&\;=&\;5\\3x&\;+&\;5y&\;+&\;6z&\;=&\;7\\2x&\;+&\;4y&\;+&\;3z&\;=&\;8\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89119b2cb41304a8c1a1f087667f76e738f8d01e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:18.915ex; height:8.843ex;" alt="{\displaystyle {\begin{alignedat}{7}x&\;+&\;3y&\;-&\;2z&\;=&\;5\\3x&\;+&\;5y&\;+&\;6z&\;=&\;7\\2x&\;+&\;4y&\;+&\;3z&\;=&\;8\end{alignedat}}}"></span></dd></dl> <p>is given by </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 {\,{\begin{vmatrix}5&3&-2\\7&5&6\\8&4&3\end{vmatrix}}\,}{\,{\begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix}}\,}},\;\;\;\;y={\frac {\,{\begin{vmatrix}1&5&-2\\3&7&6\\2&8&3\end{vmatrix}}\,}{\,{\begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix}}\,}},\;\;\;\;z={\frac {\,{\begin{vmatrix}1&3&5\\3&5&7\\2&4&8\end{vmatrix}}\,}{\,{\begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix}}\,}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>5</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>7</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mspace width="thinmathspace" /> </mrow> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mspace width="thinmathspace" /> </mrow> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>7</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mspace width="thinmathspace" /> </mrow> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {\,{\begin{vmatrix}5&3&-2\\7&5&6\\8&4&3\end{vmatrix}}\,}{\,{\begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix}}\,}},\;\;\;\;y={\frac {\,{\begin{vmatrix}1&5&-2\\3&7&6\\2&8&3\end{vmatrix}}\,}{\,{\begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix}}\,}},\;\;\;\;z={\frac {\,{\begin{vmatrix}1&3&5\\3&5&7\\2&4&8\end{vmatrix}}\,}{\,{\begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix}}\,}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eccc2f94350f08e89687a114d42682eb3bde8453" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.171ex; width:61.533ex; height:19.509ex;" alt="{\displaystyle x={\frac {\,{\begin{vmatrix}5&3&-2\\7&5&6\\8&4&3\end{vmatrix}}\,}{\,{\begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix}}\,}},\;\;\;\;y={\frac {\,{\begin{vmatrix}1&5&-2\\3&7&6\\2&8&3\end{vmatrix}}\,}{\,{\begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix}}\,}},\;\;\;\;z={\frac {\,{\begin{vmatrix}1&3&5\\3&5&7\\2&4&8\end{vmatrix}}\,}{\,{\begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix}}\,}}.}"></span></dd></dl> <p>For each variable, the denominator is the determinant of the <a href="/wiki/Matrix_of_coefficients" class="mw-redirect" title="Matrix of coefficients">matrix of coefficients</a>, while the numerator is the determinant of a matrix in which one column has been replaced by the vector of constant terms. </p><p>Though Cramer's rule is important theoretically, it has little practical value for large matrices, since the computation of large determinants is somewhat cumbersome. (Indeed, large determinants are most easily computed using row reduction.) Further, Cramer's rule has very poor numerical properties, making it unsuitable for solving even small systems reliably, unless the operations are performed in rational arithmetic with unbounded precision.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2017)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Matrix_solution">Matrix solution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=19" title="Edit section: Matrix solution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the equation system is expressed in the matrix 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\mathbf {x} =\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {x} =\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d894430af69e29d6dda5aacbf4bb19336226a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.738ex; height:2.176ex;" alt="{\displaystyle A\mathbf {x} =\mathbf {b} }"></span>, the entire solution set can also be expressed in matrix form. If the matrix <i>A</i> is square (has <i>m</i> rows and <i>n</i>=<i>m</i> columns) and has full rank (all <i>m</i> rows are independent), then the system has a unique solution given by </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 \mathbf {x} =A^{-1}\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =A^{-1}\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c663a816e9df20500c57fad3e135949594e2f95e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.071ex; height:2.676ex;" alt="{\displaystyle \mathbf {x} =A^{-1}\mathbf {b} }"></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 A^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ba3a7118652cffd5de466dc439ee9184371d50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.076ex; height:2.676ex;" alt="{\displaystyle A^{-1}}"></span> is the <a href="/wiki/Matrix_inverse" class="mw-redirect" title="Matrix inverse">inverse</a> of <i>A</i>. More generally, regardless of whether <i>m</i>=<i>n</i> or not and regardless of the rank of <i>A</i>, all solutions (if any exist) are given using the <a href="/wiki/Moore%E2%80%93Penrose_inverse" title="Moore–Penrose inverse">Moore–Penrose inverse</a> of <i>A</i>, denoted <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^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b380a5ff4e2d7d22a0dc1aea46e7ecba61f95fe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.254ex; height:2.509ex;" alt="{\displaystyle A^{+}}"></span>, as follows: </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 \mathbf {x} =A^{+}\mathbf {b} +\left(I-A^{+}A\right)\mathbf {w} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>I</mi> <mo>−</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =A^{+}\mathbf {b} +\left(I-A^{+}A\right)\mathbf {w} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dfe07d169a73899ce2ed3656f10e3cbf4bec70b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.546ex; height:3.176ex;" alt="{\displaystyle \mathbf {x} =A^{+}\mathbf {b} +\left(I-A^{+}A\right)\mathbf {w} }"></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 \mathbf {w} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20795664b5b048744a2fd88977851104cc5816f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:1.676ex;" alt="{\displaystyle \mathbf {w} }"></span> is a vector of free parameters that ranges over all possible <i>n</i>×1 vectors. A necessary and sufficient condition for any solution(s) to exist is that the potential solution obtained using <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 \mathbf {w} =\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} =\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbcfe155e035c1c6fed6f010768dda152ea4d465" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.366ex; height:2.176ex;" alt="{\displaystyle \mathbf {w} =\mathbf {0} }"></span> satisfy <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\mathbf {x} =\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {x} =\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d894430af69e29d6dda5aacbf4bb19336226a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.738ex; height:2.176ex;" alt="{\displaystyle A\mathbf {x} =\mathbf {b} }"></span> — that is, 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 AA^{+}\mathbf {b} =\mathbf {b} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AA^{+}\mathbf {b} =\mathbf {b} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a027e6b7082bf0bce3d14a468147fd873993ea9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.713ex; height:2.509ex;" alt="{\displaystyle AA^{+}\mathbf {b} =\mathbf {b} .}"></span> If this condition does not hold, the equation system is inconsistent and has no solution. If the condition holds, the system is consistent and at least one solution exists. For example, in the above-mentioned case in which <i>A</i> is square and of full rank, <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^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b380a5ff4e2d7d22a0dc1aea46e7ecba61f95fe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.254ex; height:2.509ex;" alt="{\displaystyle A^{+}}"></span> simply equals <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^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ba3a7118652cffd5de466dc439ee9184371d50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.076ex; height:2.676ex;" alt="{\displaystyle A^{-1}}"></span> and the general solution equation 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 \mathbf {x} =A^{-1}\mathbf {b} +\left(I-A^{-1}A\right)\mathbf {w} =A^{-1}\mathbf {b} +\left(I-I\right)\mathbf {w} =A^{-1}\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>I</mi> <mo>−</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>I</mi> <mo>−</mo> <mi>I</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =A^{-1}\mathbf {b} +\left(I-A^{-1}A\right)\mathbf {w} =A^{-1}\mathbf {b} +\left(I-I\right)\mathbf {w} =A^{-1}\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd1cd1f53207922a024b8ba8c8b6aa1b8426f237" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:56.662ex; height:3.343ex;" alt="{\displaystyle \mathbf {x} =A^{-1}\mathbf {b} +\left(I-A^{-1}A\right)\mathbf {w} =A^{-1}\mathbf {b} +\left(I-I\right)\mathbf {w} =A^{-1}\mathbf {b} }"></span></dd></dl> <p>as previously stated, 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 \mathbf {w} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20795664b5b048744a2fd88977851104cc5816f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:1.676ex;" alt="{\displaystyle \mathbf {w} }"></span> has completely dropped out of the solution, leaving only a single solution. In other cases, though, <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 \mathbf {w} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20795664b5b048744a2fd88977851104cc5816f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:1.676ex;" alt="{\displaystyle \mathbf {w} }"></span> remains and hence an infinitude of potential values of the free parameter vector <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 \mathbf {w} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20795664b5b048744a2fd88977851104cc5816f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:1.676ex;" alt="{\displaystyle \mathbf {w} }"></span> give an infinitude of solutions of the equation. </p> <div class="mw-heading mw-heading3"><h3 id="Other_methods">Other methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=20" title="Edit section: Other methods"><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">Further information: <a href="/wiki/Numerical_solution_of_linear_systems" class="mw-redirect" title="Numerical solution of linear systems">Numerical solution of linear systems</a></div> <p>While systems of three or four equations can be readily solved by hand (see <a href="/wiki/Cracovian" title="Cracovian">Cracovian</a>), computers are often used for larger systems. The standard algorithm for solving a system of linear equations is based on Gaussian elimination with some modifications. Firstly, it is essential to avoid division by small numbers, which may lead to inaccurate results. This can be done by reordering the equations if necessary, a process known as <a href="/wiki/Pivot_element" title="Pivot element"><i>pivoting</i></a>. Secondly, the algorithm does not exactly do Gaussian elimination, but it computes the <a href="/wiki/LU_decomposition" title="LU decomposition">LU decomposition</a> of the matrix <i>A</i>. This is mostly an organizational tool, but it is much quicker if one has to solve several systems with the same matrix <i>A</i> but different vectors <b>b</b>. </p><p>If the matrix <i>A</i> has some special structure, this can be exploited to obtain faster or more accurate algorithms. For instance, systems with a <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a> <a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive definite</a> matrix can be solved twice as fast with the <a href="/wiki/Cholesky_decomposition" title="Cholesky decomposition">Cholesky decomposition</a>. <a href="/wiki/Levinson_recursion" title="Levinson recursion">Levinson recursion</a> is a fast method for <a href="/wiki/Toeplitz_matrix" title="Toeplitz matrix">Toeplitz matrices</a>. Special methods exist also for matrices with many zero elements (so-called <a href="/wiki/Sparse_matrix" title="Sparse matrix">sparse matrices</a>), which appear often in applications. </p><p>A completely different approach is often taken for very large systems, which would otherwise take too much time or memory. The idea is to start with an initial approximation to the solution (which does not have to be accurate at all), and to change this approximation in several steps to bring it closer to the true solution. Once the approximation is sufficiently accurate, this is taken to be the solution to the system. This leads to the class of <a href="/wiki/Iterative_method" title="Iterative method">iterative methods</a>. For some sparse matrices, the introduction of randomness improves the speed of the iterative methods.<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> One example of an iterative method is the <a href="/wiki/Jacobi_method" title="Jacobi method">Jacobi method</a>, where the matrix <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is split into its diagonal component <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 D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> and its non-diagonal component <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 L+U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>+</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L+U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f553df6d71bde4d8c86baa5ffe6705a412e86e95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.206ex; height:2.343ex;" alt="{\displaystyle L+U}"></span>. An initial guess <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 {\mathbf {x}}^{(0)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {x}}^{(0)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6ebbfa25788095285f0675b14021cdb6b95ea6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.745ex; height:2.843ex;" alt="{\displaystyle {\mathbf {x}}^{(0)}}"></span> is used at the start of the algorithm. Each subsequent guess is computed using the iterative 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 {\mathbf {x}}^{(k+1)}=D^{-1}({\mathbf {b}}-(L+U){\mathbf {x}}^{(k)})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> <mo>−</mo> <mo stretchy="false">(</mo> <mi>L</mi> <mo>+</mo> <mi>U</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {x}}^{(k+1)}=D^{-1}({\mathbf {b}}-(L+U){\mathbf {x}}^{(k)})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3368551e193db8a81f92ad4fc7029600b47f35b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.164ex; height:3.343ex;" alt="{\displaystyle {\mathbf {x}}^{(k+1)}=D^{-1}({\mathbf {b}}-(L+U){\mathbf {x}}^{(k)})}"></span></dd></dl> <p>When the difference between guesses <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 {\mathbf {x}}^{(k)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {x}}^{(k)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/690e84b53186e51bc6432fb1024e0c3a3b6e5041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.779ex; height:2.843ex;" alt="{\displaystyle {\mathbf {x}}^{(k)}}"></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 {\mathbf {x}}^{(k+1)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {x}}^{(k+1)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/832f466e73981a249e41975aa77919734cfedd9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.88ex; height:2.843ex;" alt="{\displaystyle {\mathbf {x}}^{(k+1)}}"></span> is sufficiently small, the algorithm is said to have <i>converged</i> on the solution.<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>There is also a <a href="/wiki/Quantum_algorithm_for_linear_systems_of_equations" class="mw-redirect" title="Quantum algorithm for linear systems of equations">quantum algorithm for linear systems of equations</a>.<sup id="cite_ref-FOOTNOTEHarrowHassidimLloyd2009_12-0" class="reference"><a href="#cite_note-FOOTNOTEHarrowHassidimLloyd2009-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Homogeneous_systems">Homogeneous systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=21" title="Edit section: Homogeneous systems"><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/Homogeneous_differential_equation" title="Homogeneous differential equation">Homogeneous differential equation</a></div> <p>A system of linear equations is <b>homogeneous</b> if all of the constant terms are zero: </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{alignedat}{7}a_{11}x_{1}&&\;+\;&&a_{12}x_{2}&&\;+\cdots +\;&&a_{1n}x_{n}&&\;=\;&&&0\\a_{21}x_{1}&&\;+\;&&a_{22}x_{2}&&\;+\cdots +\;&&a_{2n}x_{n}&&\;=\;&&&0\\&&&&&&&&&&\vdots \;\ &&&\\a_{m1}x_{1}&&\;+\;&&a_{m2}x_{2}&&\;+\cdots +\;&&a_{mn}x_{n}&&\;=\;&&&0.\\\end{alignedat}}}"> <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 right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd /> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd /> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd> <mo>⋮</mo> <mspace width="thickmathspace" /> <mtext> </mtext> </mtd> <mtd /> <mtd /> <mtd /> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd /> <mtd> <mn>0.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{7}a_{11}x_{1}&&\;+\;&&a_{12}x_{2}&&\;+\cdots +\;&&a_{1n}x_{n}&&\;=\;&&&0\\a_{21}x_{1}&&\;+\;&&a_{22}x_{2}&&\;+\cdots +\;&&a_{2n}x_{n}&&\;=\;&&&0\\&&&&&&&&&&\vdots \;\ &&&\\a_{m1}x_{1}&&\;+\;&&a_{m2}x_{2}&&\;+\cdots +\;&&a_{mn}x_{n}&&\;=\;&&&0.\\\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4202200b0d227a22b89a37a24c8f41c521644006" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:35.951ex; height:13.176ex;" alt="{\displaystyle {\begin{alignedat}{7}a_{11}x_{1}&&\;+\;&&a_{12}x_{2}&&\;+\cdots +\;&&a_{1n}x_{n}&&\;=\;&&&0\\a_{21}x_{1}&&\;+\;&&a_{22}x_{2}&&\;+\cdots +\;&&a_{2n}x_{n}&&\;=\;&&&0\\&&&&&&&&&&\vdots \;\ &&&\\a_{m1}x_{1}&&\;+\;&&a_{m2}x_{2}&&\;+\cdots +\;&&a_{mn}x_{n}&&\;=\;&&&0.\\\end{alignedat}}}"></span></dd></dl> <p>A homogeneous system is equivalent to a matrix equation of the 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 A\mathbf {x} =\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {x} =\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f51dab2d10f98cebc72b24960edf3e14e06c1e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.589ex; height:2.176ex;" alt="{\displaystyle A\mathbf {x} =\mathbf {0} }"></span></dd></dl> <p>where <i>A</i> is an <span class="nowrap"><i>m</i> × <i>n</i></span> matrix, <b>x</b> is a column vector with <i>n</i> entries, and <b>0</b> is the <a href="/wiki/Zero_vector" class="mw-redirect" title="Zero vector">zero vector</a> with <i>m</i> entries. </p> <div class="mw-heading mw-heading3"><h3 id="Homogeneous_solution_set">Homogeneous solution set</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=22" title="Edit section: Homogeneous solution set"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every homogeneous system has at least one solution, known as the <i>zero</i> (or <i>trivial</i>) solution, which is obtained by assigning the value of zero to each of the variables. If the system has a <a href="/wiki/Non-singular_matrix" class="mw-redirect" title="Non-singular matrix">non-singular matrix</a> (<span class="texhtml">det(<i>A</i>) ≠ 0</span>) then it is also the only solution. If the system has a singular matrix then there is a solution set with an infinite number of solutions. This solution set has the following additional properties: </p> <ol><li>If <b>u</b> and <b>v</b> are two <a href="/wiki/Vector_(mathematics)" class="mw-redirect" title="Vector (mathematics)">vectors</a> representing solutions to a homogeneous system, then the vector sum <span class="nowrap"><b>u</b> + <b>v</b></span> is also a solution to the system.</li> <li>If <b>u</b> is a vector representing a solution to a homogeneous system, and <i>r</i> is any <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a>, then <i>r</i><b>u</b> is also a solution to the system.</li></ol> <p>These are exactly the properties required for the solution set to be a <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> of <b>R</b><sup><i>n</i></sup>. In particular, the solution set to a homogeneous system is the same as the <a href="/wiki/Kernel_(matrix)" class="mw-redirect" title="Kernel (matrix)">null space</a> of the corresponding matrix <i>A</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Relation_to_nonhomogeneous_systems">Relation to nonhomogeneous systems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=23" title="Edit section: Relation to nonhomogeneous systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is a close relationship between the solutions to a linear system and the solutions to the corresponding homogeneous 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 A\mathbf {x} =\mathbf {b} \qquad {\text{and}}\qquad A\mathbf {x} =\mathbf {0} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="2em" /> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {x} =\mathbf {b} \qquad {\text{and}}\qquad A\mathbf {x} =\mathbf {0} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3144f044af31ffa73f20ff753c01b7b33de8b794" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:29.012ex; height:2.176ex;" alt="{\displaystyle A\mathbf {x} =\mathbf {b} \qquad {\text{and}}\qquad A\mathbf {x} =\mathbf {0} .}"></span></dd></dl> <p>Specifically, if <b>p</b> is any specific solution to the linear system <span class="nowrap"><i>A</i><b>x</b> = <b>b</b></span>, then the entire solution set can be described 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 \left\{\mathbf {p} +\mathbf {v} :\mathbf {v} {\text{ is any solution to }}A\mathbf {x} =\mathbf {0} \right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> is any solution to </mtext> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\mathbf {p} +\mathbf {v} :\mathbf {v} {\text{ is any solution to }}A\mathbf {x} =\mathbf {0} \right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/489fd9da86ec847951c08469aafb8b4dd7adf9d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.273ex; height:2.843ex;" alt="{\displaystyle \left\{\mathbf {p} +\mathbf {v} :\mathbf {v} {\text{ is any solution to }}A\mathbf {x} =\mathbf {0} \right\}.}"></span></dd></dl> <p>Geometrically, this says that the solution set for <span class="nowrap"><i>A</i><b>x</b> = <b>b</b></span> is a <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a> of the solution set for <span class="nowrap"><i>A</i><b>x</b> = <b>0</b></span>. Specifically, the <a href="/wiki/Flat_(geometry)" title="Flat (geometry)">flat</a> for the first system can be obtained by translating the <a href="/wiki/Euclidean_subspace" class="mw-redirect" title="Euclidean subspace">linear subspace</a> for the homogeneous system by the vector <b>p</b>. </p><p>This reasoning only applies if the system <span class="nowrap"><i>A</i><b>x</b> = <b>b</b></span> has at least one solution. This occurs if and only if the vector <b>b</b> lies in the <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a> of the <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a> <i>A</i>. </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=System_of_linear_equations&action=edit&section=24" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Arrangement_of_hyperplanes" title="Arrangement of hyperplanes">Arrangement of hyperplanes</a></li> <li><a href="/wiki/Iterative_refinement" title="Iterative refinement">Iterative refinement</a> – Method to improve accuracy of numerical solutions to systems of linear equations</li> <li><a href="/wiki/Coates_graph" title="Coates graph">Coates graph</a> – A mathematical graph for solution of linear equations</li> <li><a href="/wiki/LAPACK" title="LAPACK">LAPACK</a> – Software library for numerical linear algebra</li> <li><a href="/wiki/Linear_equation_over_a_ring" title="Linear equation over a ring">Linear equation over a ring</a></li> <li><a href="/wiki/Linear_least_squares_(mathematics)" class="mw-redirect" title="Linear least squares (mathematics)">Linear least squares</a> – Least squares approximation of linear functions to data<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Matrix_decomposition" title="Matrix decomposition">Matrix decomposition</a> – Representation of a matrix as a product</li> <li><a href="/wiki/Matrix_splitting" title="Matrix splitting">Matrix splitting</a> – Representation of a matrix as a sum</li> <li><a href="/wiki/NAG_Numerical_Library" title="NAG Numerical Library">NAG Numerical Library</a> – Software library of numerical-analysis algorithms</li> <li><a href="/wiki/Rybicki_Press_algorithm" title="Rybicki Press algorithm">Rybicki Press algorithm</a> – An algorithm for inverting a matrix</li> <li><a href="/wiki/Simultaneous_equations" class="mw-redirect" title="Simultaneous equations">Simultaneous equations</a> – Set of equations to be solved together<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></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=System_of_linear_equations&action=edit&section=25" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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-FOOTNOTEAnton19872BurdenFaires1993324GolubVan_Loan199687Harper197657-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEAnton19872BurdenFaires1993324GolubVan_Loan199687Harper197657_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFAnton1987">Anton (1987)</a>, p. 2; <a href="#CITEREFBurdenFaires1993">Burden & Faires (1993)</a>, p. 324; <a href="#CITEREFGolubVan_Loan1996">Golub & Van Loan (1996)</a>, p. 87; <a href="#CITEREFHarper1976">Harper (1976)</a>, p. 57.</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"><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><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://www.britannica.com/science/system-of-equations">"System of Equations"</a>. <i>Britannica</i><span class="reference-accessdate">. Retrieved <span class="nowrap">August 26,</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=System+of+Equations&rft.btitle=Britannica&rft_id=https%3A%2F%2Fwww.britannica.com%2Fscience%2Fsystem-of-equations&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEBeauregardFraleigh197365-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBeauregardFraleigh197365_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBeauregardFraleigh1973">Beauregard & Fraleigh (1973)</a>, p. 65.</span> </li> <li id="cite_note-FOOTNOTEBeauregardFraleigh197365–66-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBeauregardFraleigh197365–66_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBeauregardFraleigh1973">Beauregard & Fraleigh (1973)</a>, pp. 65–66.</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://math.berkeley.edu/~arash/54/notes/01_01.pdf">"Systems of Linear Equations"</a> <span class="cs1-format">(PDF)</span>. <i>math.berkeley.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">February 3,</span> 2025</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=math.berkeley.edu&rft.atitle=Systems+of+Linear+Equations&rft_id=https%3A%2F%2Fmath.berkeley.edu%2F~arash%2F54%2Fnotes%2F01_01.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTECullen19903-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECullen19903_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCullen1990">Cullen (1990)</a>, p. 3.</span> </li> <li id="cite_note-FOOTNOTEWhitelaw1991[httpsbooksgooglecombooksid6M_kDzA7-qICpgPA70_70]-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWhitelaw1991[httpsbooksgooglecombooksid6M_kDzA7-qICpgPA70_70]_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWhitelaw1991">Whitelaw (1991)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6M_kDzA7-qIC&pg=PA70">70</a>.</span> </li> <li id="cite_note-FOOTNOTEBeauregardFraleigh197368-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEBeauregardFraleigh197368_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEBeauregardFraleigh197368_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFBeauregardFraleigh1973">Beauregard & Fraleigh (1973)</a>, p. 68.</span> </li> <li id="cite_note-FOOTNOTESterling2009[httpsbooksgooglecombooksidPsNJ1alC-bsCpgPA235_235]-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESterling2009[httpsbooksgooglecombooksidPsNJ1alC-bsCpgPA235_235]_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSterling2009">Sterling (2009)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PsNJ1alC-bsC&pg=PA235">235</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartnett2021" class="citation news cs1">Hartnett, Kevin (March 8, 2021). <a rel="nofollow" class="external text" href="https://www.quantamagazine.org/new-algorithm-breaks-speed-limit-for-solving-linear-equations-20210308/">"New Algorithm Breaks Speed Limit for Solving Linear Equations"</a>. <i><a href="/wiki/Quanta_Magazine" title="Quanta Magazine">Quanta Magazine</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">March 9,</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Quanta+Magazine&rft.atitle=New+Algorithm+Breaks+Speed+Limit+for+Solving+Linear+Equations&rft.date=2021-03-08&rft.aulast=Hartnett&rft.aufirst=Kevin&rft_id=https%3A%2F%2Fwww.quantamagazine.org%2Fnew-algorithm-breaks-speed-limit-for-solving-linear-equations-20210308%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/JacobiMethod.html">"Jacobi Method"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Jacobi+Method&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FJacobiMethod.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEHarrowHassidimLloyd2009-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHarrowHassidimLloyd2009_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHarrowHassidimLloyd2009">Harrow, Hassidim & Lloyd (2009)</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=26" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnton1987" class="citation cs2">Anton, Howard (1987), <i>Elementary Linear Algebra</i> (5th ed.), New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">Wiley</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-84819-0" title="Special:BookSources/0-471-84819-0"><bdi>0-471-84819-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Linear+Algebra&rft.place=New+York&rft.edition=5th&rft.pub=Wiley&rft.date=1987&rft.isbn=0-471-84819-0&rft.aulast=Anton&rft.aufirst=Howard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeauregardFraleigh1973" class="citation cs2">Beauregard, Raymond A.; Fraleigh, John B. (1973), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/firstcourseinlin0000beau"><i>A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields</i></a></span>, Boston: <a href="/wiki/Houghton_Mifflin_Company" class="mw-redirect" title="Houghton Mifflin Company">Houghton Mifflin Company</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-395-14017-X" title="Special:BookSources/0-395-14017-X"><bdi>0-395-14017-X</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+In+Linear+Algebra%3A+with+Optional+Introduction+to+Groups%2C+Rings%2C+and+Fields&rft.place=Boston&rft.pub=Houghton+Mifflin+Company&rft.date=1973&rft.isbn=0-395-14017-X&rft.aulast=Beauregard&rft.aufirst=Raymond+A.&rft.au=Fraleigh%2C+John+B.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffirstcourseinlin0000beau&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurdenFaires1993" class="citation cs2">Burden, Richard L.; Faires, J. Douglas (1993), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/numericalanalysi00burd"><i>Numerical Analysis</i></a></span> (5th ed.), Boston: <a href="/w/index.php?title=Prindle,_Weber_and_Schmidt&action=edit&redlink=1" class="new" title="Prindle, Weber and Schmidt (page does not exist)">Prindle, Weber and Schmidt</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-93219-3" title="Special:BookSources/0-534-93219-3"><bdi>0-534-93219-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Analysis&rft.place=Boston&rft.edition=5th&rft.pub=Prindle%2C+Weber+and+Schmidt&rft.date=1993&rft.isbn=0-534-93219-3&rft.aulast=Burden&rft.aufirst=Richard+L.&rft.au=Faires%2C+J.+Douglas&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnumericalanalysi00burd&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCullen1990" class="citation cs2">Cullen, Charles G. (1990), <i>Matrices and Linear Transformations</i>, MA: Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-66328-9" title="Special:BookSources/978-0-486-66328-9"><bdi>978-0-486-66328-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrices+and+Linear+Transformations&rft.place=MA&rft.pub=Dover&rft.date=1990&rft.isbn=978-0-486-66328-9&rft.aulast=Cullen&rft.aufirst=Charles+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolubVan_Loan1996" class="citation cs2">Golub, Gene H.; Van Loan, Charles F. (1996), <i>Matrix Computations</i> (3rd ed.), Baltimore: <a href="/wiki/Johns_Hopkins_University_Press" title="Johns Hopkins University Press">Johns Hopkins University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8018-5414-8" title="Special:BookSources/0-8018-5414-8"><bdi>0-8018-5414-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+Computations&rft.place=Baltimore&rft.edition=3rd&rft.pub=Johns+Hopkins+University+Press&rft.date=1996&rft.isbn=0-8018-5414-8&rft.aulast=Golub&rft.aufirst=Gene+H.&rft.au=Van+Loan%2C+Charles+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarper1976" class="citation cs2">Harper, Charlie (1976), <i>Introduction to Mathematical Physics</i>, New Jersey: <a href="/wiki/Prentice-Hall" class="mw-redirect" title="Prentice-Hall">Prentice-Hall</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-487538-9" title="Special:BookSources/0-13-487538-9"><bdi>0-13-487538-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Mathematical+Physics&rft.place=New+Jersey&rft.pub=Prentice-Hall&rft.date=1976&rft.isbn=0-13-487538-9&rft.aulast=Harper&rft.aufirst=Charlie&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarrowHassidimLloyd2009" class="citation cs2">Harrow, Aram W.; Hassidim, Avinatan; Lloyd, Seth (2009), "Quantum Algorithm for Linear Systems of Equations", <i>Physical Review Letters</i>, <b>103</b> (15): 150502, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0811.3171">0811.3171</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009PhRvL.103o0502H">2009PhRvL.103o0502H</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.103.150502">10.1103/PhysRevLett.103.150502</a>, <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/19905613">19905613</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5187993">5187993</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Quantum+Algorithm+for+Linear+Systems+of+Equations&rft.volume=103&rft.issue=15&rft.pages=150502&rft.date=2009&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A5187993%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2009PhRvL.103o0502H&rft_id=info%3Aarxiv%2F0811.3171&rft_id=info%3Apmid%2F19905613&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.103.150502&rft.aulast=Harrow&rft.aufirst=Aram+W.&rft.au=Hassidim%2C+Avinatan&rft.au=Lloyd%2C+Seth&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSterling2009" class="citation cs2">Sterling, Mary J. (2009), <i>Linear Algebra for Dummies</i>, Indianapolis, Indiana: Wiley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-43090-3" title="Special:BookSources/978-0-470-43090-3"><bdi>978-0-470-43090-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+for+Dummies&rft.place=Indianapolis%2C+Indiana&rft.pub=Wiley&rft.date=2009&rft.isbn=978-0-470-43090-3&rft.aulast=Sterling&rft.aufirst=Mary+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhitelaw1991" class="citation cs2">Whitelaw, T. A. (1991), <i>Introduction to Linear Algebra</i> (2nd ed.), CRC Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7514-0159-5" title="Special:BookSources/0-7514-0159-5"><bdi>0-7514-0159-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Linear+Algebra&rft.edition=2nd&rft.pub=CRC+Press&rft.date=1991&rft.isbn=0-7514-0159-5&rft.aulast=Whitelaw&rft.aufirst=T.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=System_of_linear_equations&action=edit&section=27" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAxler1997" class="citation book cs1">Axler, Sheldon Jay (1997). <i>Linear Algebra Done Right</i> (2nd ed.). Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-98259-0" title="Special:BookSources/0-387-98259-0"><bdi>0-387-98259-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+Done+Right&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1997&rft.isbn=0-387-98259-0&rft.aulast=Axler&rft.aufirst=Sheldon+Jay&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLay2005" class="citation book cs1">Lay, David C. (August 22, 2005). <i>Linear Algebra and Its Applications</i> (3rd ed.). Addison Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-28713-7" title="Special:BookSources/978-0-321-28713-7"><bdi>978-0-321-28713-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+and+Its+Applications&rft.edition=3rd&rft.pub=Addison+Wesley&rft.date=2005-08-22&rft.isbn=978-0-321-28713-7&rft.aulast=Lay&rft.aufirst=David+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMeyer2001" class="citation book cs1">Meyer, Carl D. (February 15, 2001). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20010301161440/http://matrixanalysis.com/DownloadChapters.html"><i>Matrix Analysis and Applied Linear Algebra</i></a>. Society for Industrial and Applied Mathematics (SIAM). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-454-8" title="Special:BookSources/978-0-89871-454-8"><bdi>978-0-89871-454-8</bdi></a>. Archived from <a rel="nofollow" class="external text" href="http://www.matrixanalysis.com/DownloadChapters.html">the original</a> on March 1, 2001.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+Analysis+and+Applied+Linear+Algebra&rft.pub=Society+for+Industrial+and+Applied+Mathematics+%28SIAM%29&rft.date=2001-02-15&rft.isbn=978-0-89871-454-8&rft.aulast=Meyer&rft.aufirst=Carl+D.&rft_id=http%3A%2F%2Fwww.matrixanalysis.com%2FDownloadChapters.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoole2006" class="citation book cs1">Poole, David (2006). <i>Linear Algebra: A Modern Introduction</i> (2nd ed.). Brooks/Cole. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-99845-3" title="Special:BookSources/0-534-99845-3"><bdi>0-534-99845-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra%3A+A+Modern+Introduction&rft.edition=2nd&rft.pub=Brooks%2FCole&rft.date=2006&rft.isbn=0-534-99845-3&rft.aulast=Poole&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnton2005" class="citation book cs1">Anton, Howard (2005). <i>Elementary Linear Algebra (Applications Version)</i> (9th ed.). Wiley International.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Linear+Algebra+%28Applications+Version%29&rft.edition=9th&rft.pub=Wiley+International&rft.date=2005&rft.aulast=Anton&rft.aufirst=Howard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeon2006" class="citation book cs1">Leon, Steven J. (2006). <i>Linear Algebra With Applications</i> (7th ed.). Pearson Prentice Hall.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+With+Applications&rft.edition=7th&rft.pub=Pearson+Prentice+Hall&rft.date=2006&rft.aulast=Leon&rft.aufirst=Steven+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrang2005" class="citation book cs1"><a href="/wiki/Gilbert_Strang" title="Gilbert Strang">Strang, Gilbert</a> (2005). <i>Linear Algebra and Its Applications</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+and+Its+Applications&rft.date=2005&rft.aulast=Strang&rft.aufirst=Gilbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPengVempala2024" class="citation journal cs1">Peng, Richard; Vempala, Santosh S. (2024). "Solving Sparse Linear Systems Faster than Matrix Multiplication". <i>Comm. ACM</i>. <b>67</b> (7): <span class="nowrap">79–</span>86. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2007.10254">2007.10254</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F3615679">10.1145/3615679</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Comm.+ACM&rft.atitle=Solving+Sparse+Linear+Systems+Faster+than+Matrix+Multiplication&rft.volume=67&rft.issue=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E79-%3C%2Fspan%3E86&rft.date=2024&rft_id=info%3Aarxiv%2F2007.10254&rft_id=info%3Adoi%2F10.1145%2F3615679&rft.aulast=Peng&rft.aufirst=Richard&rft.au=Vempala%2C+Santosh+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASystem+of+linear+equations" class="Z3988"></span></li></ul> <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=System_of_linear_equations&action=edit&section=28" 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 href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:System_of_linear_equations" class="extiw" title="commons:Category:System of linear equations">System of linear equations</a> at Wikimedia Commons</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Linear_algebra379" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Linear_algebra" title="Template:Linear algebra"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Linear_algebra" title="Template talk:Linear algebra"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Linear_algebra" title="Special:EditPage/Template:Linear algebra"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Linear_algebra379" style="font-size:114%;margin:0 4em"><a href="/wiki/Linear_algebra" title="Linear algebra">Linear algebra</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><a href="/wiki/Outline_of_linear_algebra" title="Outline of linear algebra">Outline</a></li> <li><a href="/wiki/Glossary_of_linear_algebra" title="Glossary of linear algebra">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">Scalar</a></li> <li><a href="/wiki/Euclidean_vector" title="Euclidean vector">Vector</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li> <li><a href="/wiki/Scalar_multiplication" title="Scalar multiplication">Scalar multiplication</a></li> <li><a href="/wiki/Vector_projection" title="Vector projection">Vector projection</a></li> <li><a href="/wiki/Linear_span" title="Linear span">Linear span</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">Linear projection</a></li> <li><a href="/wiki/Linear_independence" title="Linear independence">Linear independence</a></li> <li><a href="/wiki/Linear_combination" title="Linear combination">Linear combination</a></li> <li><a href="/wiki/Multilinear_map" title="Multilinear map">Multilinear map</a></li> <li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Change_of_basis" title="Change of basis">Change of basis</a></li> <li><a href="/wiki/Row_and_column_vectors" title="Row and column vectors">Row and column vectors</a></li> <li><a href="/wiki/Row_and_column_spaces" title="Row and column spaces">Row and column spaces</a></li> <li><a href="/wiki/Kernel_(linear_algebra)" title="Kernel (linear algebra)">Kernel</a></li> <li><a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">Eigenvalues and eigenvectors</a></li> <li><a href="/wiki/Transpose" title="Transpose">Transpose</a></li> <li><a class="mw-selflink selflink">Linear equations</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Euclidean_space" title="Euclidean space"><img alt="Three dimensional Euclidean space" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/80px-Linear_subspaces_with_shading.svg.png" decoding="async" width="80" height="58" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/120px-Linear_subspaces_with_shading.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/160px-Linear_subspaces_with_shading.svg.png 2x" data-file-width="325" data-file-height="236" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrices</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Block_matrix" title="Block matrix">Block</a></li> <li><a href="/wiki/Matrix_decomposition" title="Matrix decomposition">Decomposition</a></li> <li><a href="/wiki/Invertible_matrix" title="Invertible matrix">Invertible</a></li> <li><a href="/wiki/Minor_(linear_algebra)" title="Minor (linear algebra)">Minor</a></li> <li><a href="/wiki/Matrix_multiplication" title="Matrix multiplication">Multiplication</a></li> <li><a href="/wiki/Rank_(linear_algebra)" title="Rank (linear algebra)">Rank</a></li> <li><a href="/wiki/Transformation_matrix" title="Transformation matrix">Transformation</a></li> <li><a href="/wiki/Cramer%27s_rule" title="Cramer's rule">Cramer's rule</a></li> <li><a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a></li> <li><a href="/wiki/Productive_matrix" title="Productive matrix">Productive matrix</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Bilinear_map" title="Bilinear map">Bilinear</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Orthogonality" title="Orthogonality">Orthogonality</a></li> <li><a href="/wiki/Dot_product" title="Dot product">Dot product</a></li> <li><a href="/wiki/Hadamard_product_(matrices)" title="Hadamard product (matrices)">Hadamard product</a></li> <li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product space</a></li> <li><a href="/wiki/Outer_product" title="Outer product">Outer product</a></li> <li><a href="/wiki/Kronecker_product" title="Kronecker product">Kronecker product</a></li> <li><a href="/wiki/Gram%E2%80%93Schmidt_process" title="Gram–Schmidt process">Gram–Schmidt process</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Determinant" title="Determinant">Determinant</a></li> <li><a href="/wiki/Cross_product" title="Cross product">Cross product</a></li> <li><a href="/wiki/Triple_product" title="Triple product">Triple product</a></li> <li><a href="/wiki/Seven-dimensional_cross_product" title="Seven-dimensional cross product">Seven-dimensional cross product</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a></li> <li><a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior algebra</a></li> <li><a href="/wiki/Bivector" title="Bivector">Bivector</a></li> <li><a href="/wiki/Multivector" title="Multivector">Multivector</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li> <li><a href="/wiki/Outermorphism" title="Outermorphism">Outermorphism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vector_space" title="Vector space">Vector space</a> constructions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dual_space" title="Dual space">Dual</a></li> <li><a href="/wiki/Direct_sum_of_modules#Construction_for_two_vector_spaces" title="Direct sum of modules">Direct sum</a></li> <li><a href="/wiki/Function_space#In_linear_algebra" title="Function space">Function space</a></li> <li><a href="/wiki/Quotient_space_(linear_algebra)" title="Quotient space (linear algebra)">Quotient</a></li> <li><a href="/wiki/Linear_subspace" title="Linear subspace">Subspace</a></li> <li><a href="/wiki/Tensor_product" title="Tensor product">Tensor product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Numerical_linear_algebra" title="Numerical linear algebra">Numerical</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">Floating-point</a></li> <li><a href="/wiki/Numerical_stability" title="Numerical stability">Numerical stability</a></li> <li><a href="/wiki/Basic_Linear_Algebra_Subprograms" title="Basic Linear Algebra Subprograms">Basic Linear Algebra Subprograms</a></li> <li><a href="/wiki/Sparse_matrix" title="Sparse matrix">Sparse matrix</a></li> <li><a href="/wiki/Comparison_of_linear_algebra_libraries" title="Comparison of linear algebra libraries">Comparison of linear algebra libraries</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Linear_algebra" title="Category:Linear algebra">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox389" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11203#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4035826-4">Germany</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=System_of_linear_equations&oldid=1273779114">https://en.wikipedia.org/w/index.php?title=System_of_linear_equations&oldid=1273779114</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Equations" title="Category:Equations">Equations</a></li><li><a href="/wiki/Category:Linear_algebra" title="Category:Linear algebra">Linear algebra</a></li><li><a href="/wiki/Category:Numerical_linear_algebra" title="Category:Numerical linear algebra">Numerical linear algebra</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Use_American_English_from_February_2025" title="Category:Use American English from February 2025">Use American English from February 2025</a></li><li><a href="/wiki/Category:All_Wikipedia_articles_written_in_American_English" title="Category:All Wikipedia articles written in American English">All Wikipedia articles written in American English</a></li><li><a href="/wiki/Category:Use_mdy_dates_from_February_2025" title="Category:Use mdy dates from February 2025">Use mdy dates from February 2025</a></li><li><a href="/wiki/Category:Articles_lacking_in-text_citations_from_October_2015" title="Category:Articles lacking in-text citations from October 2015">Articles lacking in-text citations from October 2015</a></li><li><a href="/wiki/Category:All_articles_lacking_in-text_citations" title="Category:All articles lacking in-text citations">All articles lacking in-text citations</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_March_2017" title="Category:Articles with unsourced statements from March 2017">Articles with unsourced statements from March 2017</a></li><li><a href="/wiki/Category:Pages_displaying_short_descriptions_of_redirect_targets_via_Module:Annotated_link" title="Category:Pages displaying short descriptions of redirect targets via Module:Annotated link">Pages displaying short descriptions of redirect targets via Module:Annotated link</a></li><li><a href="/wiki/Category:Commons_category_link_from_Wikidata" title="Category:Commons category link from Wikidata">Commons category link from Wikidata</a></li><li><a href="/wiki/Category:Pages_that_use_a_deprecated_format_of_the_math_tags" title="Category:Pages that use a deprecated format of the math tags">Pages that use a deprecated format of the math tags</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 4 February 2025, at 00:10<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=System_of_linear_equations&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" lang="en" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/w/resources/assets/poweredby_mediawiki.svg" width="88" height="31"><img src="/w/resources/assets/mediawiki_compact.svg" alt="Powered by MediaWiki" width="25" height="25" loading="lazy"></picture></a></li> </ul> </footer> </div> </div> </div> <div class="vector-header-container vector-sticky-header-container"> <div id="vector-sticky-header" class="vector-sticky-header"> <div class="vector-sticky-header-start"> <div class="vector-sticky-header-icon-start vector-button-flush-left vector-button-flush-right" aria-hidden="true"> <button class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-sticky-header-search-toggle" tabindex="-1" data-event-name="ui.vector-sticky-search-form.icon"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </button> </div> <div role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box"> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail"> <form action="/w/index.php" id="vector-sticky-search-form" class="cdx-search-input cdx-search-input--has-end-button"> <div class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia"> <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <div class="vector-sticky-header-context-bar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-sticky-header-toc" class="vector-dropdown mw-portlet mw-portlet-sticky-header-toc vector-sticky-header-toc vector-button-flush-left" > <input type="checkbox" id="vector-sticky-header-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-sticky-header-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-sticky-header-toc-label" for="vector-sticky-header-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-sticky-header-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" ><span class="mw-page-title-main">System of linear equations</span></div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>71 languages</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Add topic</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-b766959bd-bqm7f","wgBackendResponseTime":125,"wgPageParseReport":{"limitreport":{"cputime":"1.126","walltime":"1.515","ppvisitednodes":{"value":4492,"limit":1000000},"postexpandincludesize":{"value":65635,"limit":2097152},"templateargumentsize":{"value":3486,"limit":2097152},"expansiondepth":{"value":13,"limit":100},"expensivefunctioncount":{"value":10,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":75308,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 1205.182 1 -total"," 26.57% 320.215 10 Template:Annotated_link"," 12.96% 156.234 1 Template:Reflist"," 10.55% 127.137 1 Template:Short_description"," 10.13% 122.113 1 Template:Linear_algebra"," 9.89% 119.232 1 Template:Navbox"," 8.27% 99.723 1 Template:Cite_encyclopedia"," 7.45% 89.749 2 Template:Pagetype"," 6.81% 82.068 1 Template:More_footnotes"," 6.19% 74.614 1 Template:Ambox"]},"scribunto":{"limitreport-timeusage":{"value":"0.744","limit":"10.000"},"limitreport-memusage":{"value":20176719,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAnton1987\"] = 1,\n [\"CITEREFAnton2005\"] = 1,\n [\"CITEREFAxler1997\"] = 1,\n [\"CITEREFBeauregardFraleigh1973\"] = 1,\n [\"CITEREFBurdenFaires1993\"] = 1,\n [\"CITEREFCullen1990\"] = 1,\n [\"CITEREFGolubVan_Loan1996\"] = 1,\n [\"CITEREFHarper1976\"] = 1,\n [\"CITEREFHarrowHassidimLloyd2009\"] = 1,\n [\"CITEREFHartnett2021\"] = 1,\n [\"CITEREFLay2005\"] = 1,\n [\"CITEREFLeon2006\"] = 1,\n [\"CITEREFMeyer2001\"] = 1,\n [\"CITEREFPengVempala2024\"] = 1,\n [\"CITEREFPoole2006\"] = 1,\n [\"CITEREFSterling2009\"] = 1,\n [\"CITEREFStrang2005\"] = 1,\n [\"CITEREFWhitelaw1991\"] = 1,\n}\ntemplate_list = table#1 {\n [\"=\"] = 14,\n [\"Annotated link\"] = 10,\n [\"Authority control\"] = 1,\n [\"Citation\"] = 9,\n [\"Citation needed\"] = 1,\n [\"Cite book\"] = 7,\n [\"Cite encyclopedia\"] = 1,\n [\"Cite journal\"] = 1,\n [\"Cite news\"] = 1,\n [\"Cite web\"] = 2,\n [\"Commons category-inline\"] = 1,\n [\"Further\"] = 1,\n [\"Linear algebra\"] = 1,\n [\"Main\"] = 2,\n [\"Math\"] = 2,\n [\"More footnotes\"] = 1,\n [\"Nowrap\"] = 21,\n [\"Reflist\"] = 1,\n [\"See also\"] = 2,\n [\"Sfnmp\"] = 1,\n [\"Sfnp\"] = 8,\n [\"Short description\"] = 1,\n [\"Use American English\"] = 1,\n [\"Use mdy dates\"] = 1,\n}\narticle_whitelist = table#1 {\n}\nciteref_patterns = table#1 {\n}\n"},"cachereport":{"origin":"mw-api-int.codfw.main-5b65fffc7d-s8gvv","timestamp":"20250214041239","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"System of linear equations","url":"https:\/\/en.wikipedia.org\/wiki\/System_of_linear_equations","sameAs":"http:\/\/www.wikidata.org\/entity\/Q11203","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q11203","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2002-10-18T23:20:31Z","dateModified":"2025-02-04T00:10:09Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/a\/ab\/Secretsharing_3-point.svg","headline":"collection of linear equations involving the same set of variables"}</script> </body> </html>

CINXE.COM

System of linear equations - Wikipedia