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href="https://cy.wikipedia.org/wiki/Hafaliad"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.cy"> <link rel="alternate" type="application/atom+xml" title="Ffrwd Atom Wicipedia" href="/w/index.php?title=Arbennig: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-Hafaliad rootpage-Hafaliad skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Neidio i'r cynnwys</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="Safle"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <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="Prif ddewislen" > <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">Prif ddewislen</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">Prif ddewislen</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">symud i'r bar ochr</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">cuddio</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Panel llywio </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage" class="mw-list-item"><a href="/wiki/Hafan" title="Ymweld â'r Hafan [z]" accesskey="z"><span>Hafan</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wicipedia:Porth_y_Gymuned" title="Pethau i'w gwneud, adnoddau a thudalennau'r gymuned"><span>Porth y Gymuned</span></a></li><li id="n-cafe" class="mw-list-item"><a href="/wiki/Wicipedia:Y_Caffi"><span>Y Caffi</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Categori:Materion_cyfoes" title="Gwybodaeth yn gysylltiedig â materion cyfoes"><span>Materion cyfoes</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Arbennig:RecentChanges" title="Rhestr y newidiadau diweddar ar y wici. [r]" accesskey="r"><span>Newidiadau diweddar</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Arbennig:Random" title="Llwytho tudalen ar hap [x]" accesskey="x"><span>Erthygl ar hap</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Wicipedia:Cymorth" title="Tudalennau cymorth"><span>Cymorth</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Hafan" 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="Wicipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-cy.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="" src="/static/images/mobile/copyright/wikipedia-tagline-cy.svg" width="120" height="13" style="width: 7.5em; 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/Arbennig:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Chwilio Wicipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Chwilio</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="Chwilio Wicipedia" aria-label="Chwilio Wicipedia" autocapitalize="sentences" title="Chwilio Wicipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Arbennig:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Chwilio</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Offer personol"> <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="Gwedd"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Newidiwch ymddangosiad maint ffont, lled a lliw y dudalen" > <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="Gwedd" > <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">Gwedd</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="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_cy.wikipedia.org&uselang=cy" class=""><span>Rhoi</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=Arbennig:CreateAccount&returnto=Hafaliad" title="Rydym yn argymell eich bod yn creu cyfri ac yn menwgofnodi. Fodd bynnag, dydy hyn ddim yn orfodol" class=""><span>Creu cyfrif</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=Arbennig:UserLogin&returnto=Hafaliad" title="Fe'ch anogir i fewngofnodi, er nad oes rhaid gwneud. [o]" accesskey="o" class=""><span>Mewngofnodi</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="Rhagor o opsiynau" > <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="Offer personol" > <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">Offer personol</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Dewislen defnyddiwr" > <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="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_cy.wikipedia.org&uselang=cy"><span>Rhoi</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Arbennig:CreateAccount&returnto=Hafaliad" title="Rydym yn argymell eich bod yn creu cyfri ac yn menwgofnodi. Fodd bynnag, dydy hyn ddim yn orfodol"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Creu cyfrif</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Arbennig:UserLogin&returnto=Hafaliad" title="Fe'ch anogir i fewngofnodi, er nad oes rhaid gwneud. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Mewngofnodi</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"> Tudalennau ar gyfer golygyddion allgofnodedig <a href="/wiki/Cymorth:Cyflwyniad" aria-label="Dysgu mwy am olygu"><span>dysgu mwy</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/Arbennig:MyContributions" title="Rhestr golygiadau o'r cyfeiriad IP hwn [y]" accesskey="y"><span>Cyfraniadau</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Arbennig:MyTalk" title="Sgwrs ynglŷn â golygiadau o'r cyfeiriad IP hwn [n]" accesskey="n"><span>Sgwrs</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="Safle"> <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="Cynnwys" 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">Cynnwys</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">symud i'r bar ochr</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">cuddio</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">Y dechrau</div> </a> </li> <li id="toc-Priodweddau_elfennol" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Priodweddau_elfennol"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Priodweddau elfennol</span> </div> </a> <ul id="toc-Priodweddau_elfennol-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cyflwyniad" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Cyflwyniad"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Cyflwyniad</span> </div> </a> <button aria-controls="toc-Cyflwyniad-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>Toglo is-adran Cyflwyniad</span> </button> <ul id="toc-Cyflwyniad-sublist" class="vector-toc-list"> <li id="toc-Darlun_analog" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Darlun_analog"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Darlun analog</span> </div> </a> <ul id="toc-Darlun_analog-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Paramedrau_ac_anhysbysion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Paramedrau_ac_anhysbysion"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Paramedrau ac anhysbysion</span> </div> </a> <ul id="toc-Paramedrau_ac_anhysbysion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Unfathiant" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Unfathiant"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Unfathiant</span> </div> </a> <ul id="toc-Unfathiant-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Priodweddau" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Priodweddau"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Priodweddau</span> </div> </a> <ul id="toc-Priodweddau-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algebra"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Algebra</span> </div> </a> <button aria-controls="toc-Algebra-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>Toglo is-adran Algebra</span> </button> <ul id="toc-Algebra-sublist" class="vector-toc-list"> <li id="toc-Hafaliadau_polynomial" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hafaliadau_polynomial"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Hafaliadau polynomial</span> </div> </a> <ul id="toc-Hafaliadau_polynomial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Systemau_hafaliadau_llinol" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Systemau_hafaliadau_llinol"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Systemau hafaliadau llinol</span> </div> </a> <ul id="toc-Systemau_hafaliadau_llinol-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Geometreg" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geometreg"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Geometreg</span> </div> </a> <button aria-controls="toc-Geometreg-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>Toglo is-adran Geometreg</span> </button> <ul id="toc-Geometreg-sublist" class="vector-toc-list"> <li id="toc-Geometreg_ddadansoddol" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometreg_ddadansoddol"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Geometreg ddadansoddol</span> </div> </a> <ul id="toc-Geometreg_ddadansoddol-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hafaliadau_Cartesaidd" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hafaliadau_Cartesaidd"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Hafaliadau Cartesaidd</span> </div> </a> <ul id="toc-Hafaliadau_Cartesaidd-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Damcaniaeth_rhif" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Damcaniaeth_rhif"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Damcaniaeth rhif</span> </div> </a> <button aria-controls="toc-Damcaniaeth_rhif-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>Toglo is-adran Damcaniaeth rhif</span> </button> <ul id="toc-Damcaniaeth_rhif-sublist" class="vector-toc-list"> <li id="toc-Hafaliadau_Diophantine" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hafaliadau_Diophantine"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Hafaliadau Diophantine</span> </div> </a> <ul id="toc-Hafaliadau_Diophantine-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Hafaliadau_differol" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Hafaliadau_differol"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Hafaliadau differol</span> </div> </a> <button aria-controls="toc-Hafaliadau_differol-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>Toglo is-adran Hafaliadau differol</span> </button> <ul id="toc-Hafaliadau_differol-sublist" class="vector-toc-list"> <li id="toc-Geometreg_algebraidd" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometreg_algebraidd"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Geometreg algebraidd</span> </div> </a> <ul id="toc-Geometreg_algebraidd-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mathau_o_hafaliadau" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mathau_o_hafaliadau"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Mathau o hafaliadau</span> </div> </a> <ul id="toc-Mathau_o_hafaliadau-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cyfeiriadau" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Cyfeiriadau"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Cyfeiriadau</span> </div> </a> <ul id="toc-Cyfeiriadau-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dolenni_allanol" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dolenni_allanol"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Dolenni allanol</span> </div> </a> <ul id="toc-Dolenni_allanol-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="Cynnwys" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toglo'r tabl cynnwys" > <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">Toglo'r tabl cynnwys</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">Hafaliad</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="Ewch i erthygl mewn iaith arall. Ar gael mewn 122 iaith" > <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-122" 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">122 iaith</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Vergelyking_(wiskunde)" title="Vergelyking (wiskunde) - Affricaneg" lang="af" hreflang="af" data-title="Vergelyking (wiskunde)" data-language-autonym="Afrikaans" data-language-local-name="Affricaneg" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Gleichung" title="Gleichung - Almaeneg y Swistir" lang="gsw" hreflang="gsw" data-title="Gleichung" data-language-autonym="Alemannisch" data-language-local-name="Almaeneg y Swistir" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Ecuaci%C3%B3n" title="Ecuación - Aragoneg" lang="an" hreflang="an" data-title="Ecuación" data-language-autonym="Aragonés" data-language-local-name="Aragoneg" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3" title="समीकरण - Angika" lang="anp" hreflang="anp" data-title="समीकरण" data-language-autonym="अंगिका" data-language-local-name="Angika" class="interlanguage-link-target"><span>अंगिका</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A9_%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A9" title="معادلة رياضية - Arabeg" lang="ar" hreflang="ar" data-title="معادلة رياضية" data-language-autonym="العربية" data-language-local-name="Arabeg" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A9_%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A9" title="معادلة رياضية - Arabeg yr Aifft" lang="arz" hreflang="arz" data-title="معادلة رياضية" data-language-autonym="مصرى" data-language-local-name="Arabeg yr Aifft" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%B8%E0%A6%AE%E0%A7%80%E0%A6%95%E0%A7%B0%E0%A6%A3" title="সমীকৰণ - Asameg" lang="as" hreflang="as" data-title="সমীকৰণ" data-language-autonym="অসমীয়া" data-language-local-name="Asameg" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Ecuaci%C3%B3n" title="Ecuación - Astwrianeg" lang="ast" hreflang="ast" data-title="Ecuación" data-language-autonym="Asturianu" data-language-local-name="Astwrianeg" 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/T%C9%99nlik" title="Tənlik - Aserbaijaneg" lang="az" hreflang="az" data-title="Tənlik" data-language-autonym="Azərbaycanca" data-language-local-name="Aserbaijaneg" 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/%D0%A2%D0%B8%D0%B3%D0%B5%D2%99%D0%BB%D3%99%D0%BC%D3%99" title="Тигеҙләмә - Bashcorteg" lang="ba" hreflang="ba" data-title="Тигеҙләмә" data-language-autonym="Башҡортса" data-language-local-name="Bashcorteg" 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%A3%D1%80%D0%B0%D1%9E%D0%BD%D0%B5%D0%BD%D0%BD%D0%B5" title="Ураўненне - Belarwseg" lang="be" hreflang="be" data-title="Ураўненне" data-language-autonym="Беларуская" data-language-local-name="Belarwseg" 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%A0%D0%B0%D1%9E%D0%BD%D0%B0%D0%BD%D1%8C%D0%BD%D0%B5" title="Раўнаньне - Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Раўнаньне" 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%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5" title="Уравнение - Bwlgareg" lang="bg" hreflang="bg" data-title="Уравнение" data-language-autonym="Български" data-language-local-name="Bwlgareg" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%AE%E0%A7%80%E0%A6%95%E0%A6%B0%E0%A6%A3" title="সমীকরণ - Bengaleg" lang="bn" hreflang="bn" data-title="সমীকরণ" data-language-autonym="বাংলা" data-language-local-name="Bengaleg" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Jedna%C4%8Dina" title="Jednačina - Bosnieg" lang="bs" hreflang="bs" data-title="Jednačina" data-language-autonym="Bosanski" data-language-local-name="Bosnieg" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%A2%D1%8D%D0%B3%D1%88%D1%8D%D0%B4%D1%85%D1%8D%D0%BB" title="Тэгшэдхэл - Russia Buriat" lang="bxr" hreflang="bxr" data-title="Тэгшэдхэл" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Equaci%C3%B3" title="Equació - Catalaneg" lang="ca" hreflang="ca" data-title="Equació" data-language-autonym="Català" data-language-local-name="Catalaneg" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-chr mw-list-item"><a href="https://chr.wikipedia.org/wiki/%E1%8E%A2%E1%8F%97%E1%8E%A6%E1%8F%B2%E1%8F%8D%E1%8F%97" title="ᎢᏗᎦᏲᏍᏗ - Tsierocî" lang="chr" hreflang="chr" data-title="ᎢᏗᎦᏲᏍᏗ" data-language-autonym="ᏣᎳᎩ" data-language-local-name="Tsierocî" class="interlanguage-link-target"><span>ᏣᎳᎩ</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%BE%D8%A7%D9%88%DA%A9%DB%8E%D8%B4%DB%95" title="ھاوکێشە - Cwrdeg Sorani" lang="ckb" hreflang="ckb" data-title="ھاوکێشە" data-language-autonym="کوردی" data-language-local-name="Cwrdeg Sorani" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Rovnice" title="Rovnice - Tsieceg" lang="cs" hreflang="cs" data-title="Rovnice" data-language-autonym="Čeština" data-language-local-name="Tsieceg" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%D0%B0%D0%BD%D0%BB%C4%83%D1%85_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Танлăх (математика) - Tshwfasheg" lang="cv" hreflang="cv" data-title="Танлăх (математика)" data-language-autonym="Чӑвашла" data-language-local-name="Tshwfasheg" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Ligning" title="Ligning - Daneg" lang="da" hreflang="da" data-title="Ligning" data-language-autonym="Dansk" data-language-local-name="Daneg" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gleichung" title="Gleichung - Almaeneg" lang="de" hreflang="de" data-title="Gleichung" data-language-autonym="Deutsch" data-language-local-name="Almaeneg" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CE%BE%CE%AF%CF%83%CF%89%CF%83%CE%B7" title="Εξίσωση - Groeg" lang="el" hreflang="el" data-title="Εξίσωση" data-language-autonym="Ελληνικά" data-language-local-name="Groeg" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/Equazi%C3%A5n" title="Equaziån - Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Equaziån" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Equation" title="Equation - Saesneg" lang="en" hreflang="en" data-title="Equation" data-language-autonym="English" data-language-local-name="Saesneg" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Ekvacio" title="Ekvacio - Esperanto" lang="eo" hreflang="eo" data-title="Ekvacio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ecuaci%C3%B3n" title="Ecuación - Sbaeneg" lang="es" hreflang="es" data-title="Ecuación" data-language-autonym="Español" data-language-local-name="Sbaeneg" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/V%C3%B5rrand" title="Võrrand - Estoneg" lang="et" hreflang="et" data-title="Võrrand" data-language-autonym="Eesti" data-language-local-name="Estoneg" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Ekuazio" title="Ekuazio - Basgeg" lang="eu" hreflang="eu" data-title="Ekuazio" data-language-autonym="Euskara" data-language-local-name="Basgeg" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/Ecuaci%C3%B3n" title="Ecuación - Extremadureg" lang="ext" hreflang="ext" data-title="Ecuación" data-language-autonym="Estremeñu" data-language-local-name="Extremadureg" class="interlanguage-link-target"><span>Estremeñu</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D9%87" title="معادله - Perseg" lang="fa" hreflang="fa" data-title="معادله" data-language-autonym="فارسی" data-language-local-name="Perseg" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Yht%C3%A4l%C3%B6" title="Yhtälö - Ffinneg" lang="fi" hreflang="fi" data-title="Yhtälö" data-language-autonym="Suomi" data-language-local-name="Ffinneg" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/V%C3%B5rrand" title="Võrrand - Võro" lang="vro" hreflang="vro" data-title="Võrrand" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Ivakatakata" title="Ivakatakata - Ffijïeg" lang="fj" hreflang="fj" data-title="Ivakatakata" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Ffijïeg" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/L%C3%ADkning" title="Líkning - Ffaröeg" lang="fo" hreflang="fo" data-title="Líkning" data-language-autonym="Føroyskt" data-language-local-name="Ffaröeg" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr badge-Q17437796 badge-featuredarticle mw-list-item" title="erthygl dan sylw"><a href="https://fr.wikipedia.org/wiki/%C3%89quation" title="Équation - Ffrangeg" lang="fr" hreflang="fr" data-title="Équation" data-language-autonym="Français" data-language-local-name="Ffrangeg" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Liknang" title="Liknang - Ffriseg Gogleddol" lang="frr" hreflang="frr" data-title="Liknang" data-language-autonym="Nordfriisk" data-language-local-name="Ffriseg Gogleddol" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Cothrom%C3%B3id" title="Cothromóid - Gwyddeleg" lang="ga" hreflang="ga" data-title="Cothromóid" data-language-autonym="Gaeilge" data-language-local-name="Gwyddeleg" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E6%96%B9%E7%A8%8B" title="方程 - Gan" lang="gan" hreflang="gan" data-title="方程" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/%C3%89kwasyon" title="Ékwasyon - Guianan Creole" lang="gcr" hreflang="gcr" data-title="Ékwasyon" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Ecuaci%C3%B3n" title="Ecuación - Galisieg" lang="gl" hreflang="gl" data-title="Ecuación" data-language-autonym="Galego" data-language-local-name="Galisieg" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%94" title="משוואה - Hebraeg" lang="he" hreflang="he" data-title="משוואה" data-language-autonym="עברית" data-language-local-name="Hebraeg" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3" title="समीकरण - Hindi" lang="hi" hreflang="hi" data-title="समीकरण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Baraabri" title="Baraabri - Fiji Hindi" lang="hif" hreflang="hif" data-title="Baraabri" 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-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Jednad%C5%BEba" title="Jednadžba - Croateg" lang="hr" hreflang="hr" data-title="Jednadžba" data-language-autonym="Hrvatski" data-language-local-name="Croateg" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Egyenlet" title="Egyenlet - Hwngareg" lang="hu" hreflang="hu" data-title="Egyenlet" data-language-autonym="Magyar" data-language-local-name="Hwngareg" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D5%BE%D5%A1%D5%BD%D5%A1%D6%80%D5%B8%D6%82%D5%B4" title="Հավասարում - Armeneg" lang="hy" hreflang="hy" data-title="Հավասարում" data-language-autonym="Հայերեն" data-language-local-name="Armeneg" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Equation" title="Equation - Interlingua" lang="ia" hreflang="ia" data-title="Equation" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Persamaan" title="Persamaan - Indoneseg" lang="id" hreflang="id" data-title="Persamaan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indoneseg" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Equaciono" title="Equaciono - Ido" lang="io" hreflang="io" data-title="Equaciono" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Jafna" title="Jafna - Islandeg" lang="is" hreflang="is" data-title="Jafna" data-language-autonym="Íslenska" data-language-local-name="Islandeg" 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/Equazione" title="Equazione - Eidaleg" lang="it" hreflang="it" data-title="Equazione" data-language-autonym="Italiano" data-language-local-name="Eidaleg" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%96%B9%E7%A8%8B%E5%BC%8F" title="方程式 - Japaneeg" lang="ja" hreflang="ja" data-title="方程式" data-language-autonym="日本語" data-language-local-name="Japaneeg" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Ikwiejan" title="Ikwiejan - Jamaican Creole English" lang="jam" hreflang="jam" data-title="Ikwiejan" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%92%E1%83%90%E1%83%9C%E1%83%A2%E1%83%9D%E1%83%9A%E1%83%94%E1%83%91%E1%83%90" title="განტოლება - Georgeg" lang="ka" hreflang="ka" data-title="განტოლება" data-language-autonym="ქართული" data-language-local-name="Georgeg" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/K%C9%A9ma%C5%8B_w%C9%9B%CA%8A%CA%8A" title="Kɩmaŋ wɛʊʊ - Kabiye" lang="kbp" hreflang="kbp" data-title="Kɩmaŋ wɛʊʊ" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target"><span>Kabɩyɛ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D0%B5%D2%A3%D0%B4%D0%B5%D1%83" title="Теңдеу - Casacheg" lang="kk" hreflang="kk" data-title="Теңдеу" data-language-autonym="Қазақша" data-language-local-name="Casacheg" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%9F%E1%9E%98%E1%9E%B8%E1%9E%80%E1%9E%B6%E1%9E%9A" title="សមីការ - Chmereg" lang="km" hreflang="km" data-title="សមីការ" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Chmereg" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B8%E0%B2%AE%E0%B3%80%E0%B2%95%E0%B2%B0%E0%B2%A3" title="ಸಮೀಕರಣ - Kannada" lang="kn" hreflang="kn" data-title="ಸಮೀಕರಣ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B0%A9%EC%A0%95%EC%8B%9D" title="방정식 - Coreeg" lang="ko" hreflang="ko" data-title="방정식" data-language-autonym="한국어" data-language-local-name="Coreeg" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Wekhev%C3%AE" title="Wekhevî - Cwrdeg" lang="ku" hreflang="ku" data-title="Wekhevî" data-language-autonym="Kurdî" data-language-local-name="Cwrdeg" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%A1%D0%BE%D0%BC%D0%BE/%D0%A2%D0%B5%D2%A3%D0%B4%D0%B5%D0%BC%D0%B5" title="Википедия:Сомо/Теңдеме - Cirgiseg" lang="ky" hreflang="ky" data-title="Википедия:Сомо/Теңдеме" data-language-autonym="Кыргызча" data-language-local-name="Cirgiseg" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Aequatio" title="Aequatio - Lladin" lang="la" hreflang="la" data-title="Aequatio" data-language-autonym="Latina" data-language-local-name="Lladin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Egali" title="Egali - Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Egali" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Equazion" title="Equazion - Lombardeg" lang="lmo" hreflang="lmo" data-title="Equazion" data-language-autonym="Lombard" data-language-local-name="Lombardeg" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%AA%E0%BA%BB%E0%BA%A1%E0%BA%9C%E0%BA%BB%E0%BA%99" title="ສົມຜົນ - Laoeg" lang="lo" hreflang="lo" data-title="ສົມຜົນ" data-language-autonym="ລາວ" data-language-local-name="Laoeg" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Lygtis" title="Lygtis - Lithwaneg" lang="lt" hreflang="lt" data-title="Lygtis" data-language-autonym="Lietuvių" data-language-local-name="Lithwaneg" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Vien%C4%81dojums" title="Vienādojums - Latfieg" lang="lv" hreflang="lv" data-title="Vienādojums" data-language-autonym="Latviešu" data-language-local-name="Latfieg" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B2%D0%B5%D0%BD%D0%BA%D0%B0" title="Равенка - Macedoneg" lang="mk" hreflang="mk" data-title="Равенка" data-language-autonym="Македонски" data-language-local-name="Macedoneg" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B4%AE%E0%B4%B5%E0%B4%BE%E0%B4%95%E0%B5%8D%E0%B4%AF%E0%B4%82_(%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%B6%E0%B4%BE%E0%B4%B8%E0%B5%8D%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%82)" title="സമവാക്യം (ഗണിതശാസ്ത്രം) - Malayalam" lang="ml" hreflang="ml" data-title="സമവാക്യം (ഗണിതശാസ്ത്രം)" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3" title="समीकरण - Marathi" lang="mr" hreflang="mr" data-title="समीकरण" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Persamaan" title="Persamaan - Maleieg" lang="ms" hreflang="ms" data-title="Persamaan" data-language-autonym="Bahasa Melayu" data-language-local-name="Maleieg" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nap mw-list-item"><a href="https://nap.wikipedia.org/wiki/Equazzione" title="Equazzione - Naplieg" lang="nap" hreflang="nap" data-title="Equazzione" data-language-autonym="Napulitano" data-language-local-name="Naplieg" class="interlanguage-link-target"><span>Napulitano</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Glieken" title="Glieken - Almaeneg Isel" lang="nds" hreflang="nds" data-title="Glieken" data-language-autonym="Plattdüütsch" data-language-local-name="Almaeneg Isel" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Vergelijking_(wiskunde)" title="Vergelijking (wiskunde) - Iseldireg" lang="nl" hreflang="nl" data-title="Vergelijking (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Iseldireg" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Likning" title="Likning - Norwyeg Nynorsk" lang="nn" hreflang="nn" data-title="Likning" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwyeg Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Ligning_(matematikk)" title="Ligning (matematikk) - Norwyeg Bokmål" lang="nb" hreflang="nb" data-title="Ligning (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="Norwyeg Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Equacion" title="Equacion - Ocsitaneg" lang="oc" hreflang="oc" data-title="Equacion" data-language-autonym="Occitan" data-language-local-name="Ocsitaneg" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Qixxaatoo" title="Qixxaatoo - Oromo" lang="om" hreflang="om" data-title="Qixxaatoo" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A8%AE%E0%A9%80%E0%A8%95%E0%A8%B0%E0%A8%A8" title="ਸਮੀਕਰਨ - Pwnjabeg" lang="pa" hreflang="pa" data-title="ਸਮੀਕਰਨ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Pwnjabeg" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/R%C3%B3wnanie" title="Równanie - Pwyleg" lang="pl" hreflang="pl" data-title="Równanie" data-language-autonym="Polski" data-language-local-name="Pwyleg" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Equassion" title="Equassion - Piedmonteg" lang="pms" hreflang="pms" data-title="Equassion" data-language-autonym="Piemontèis" data-language-local-name="Piedmonteg" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%AA%D8%B1%DA%A9%DA%91%DB%8C" title="ترکڑی - Western Punjabi" lang="pnb" hreflang="pnb" data-title="ترکڑی" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Equa%C3%A7%C3%A3o" title="Equação - Portiwgaleg" lang="pt" hreflang="pt" data-title="Equação" data-language-autonym="Português" data-language-local-name="Portiwgaleg" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Paqtachani" title="Paqtachani - Quechua" lang="qu" hreflang="qu" data-title="Paqtachani" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Ecua%C8%9Bie" title="Ecuație - Rwmaneg" lang="ro" hreflang="ro" data-title="Ecuație" data-language-autonym="Română" data-language-local-name="Rwmaneg" 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%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5" title="Уравнение - Rwseg" lang="ru" hreflang="ru" data-title="Уравнение" data-language-autonym="Русский" data-language-local-name="Rwseg" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%A2%D1%8D%D2%A5%D0%BD%D1%8D%D0%B1%D0%B8%D0%BB" title="Тэҥнэбил - Sakha" lang="sah" hreflang="sah" data-title="Тэҥнэбил" data-language-autonym="Саха тыла" data-language-local-name="Sakha" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Iquazzioni" title="Iquazzioni - Sisileg" lang="scn" hreflang="scn" data-title="Iquazzioni" data-language-autonym="Sicilianu" data-language-local-name="Sisileg" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Equation" title="Equation - Sgoteg" lang="sco" hreflang="sco" data-title="Equation" data-language-autonym="Scots" data-language-local-name="Sgoteg" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Jedna%C4%8Dina" title="Jednačina - Serbo-Croateg" lang="sh" hreflang="sh" data-title="Jednačina" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croateg" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/Tagdazalt" title="Tagdazalt - Tachelhit" lang="shi" hreflang="shi" data-title="Tagdazalt" data-language-autonym="Taclḥit" data-language-local-name="Tachelhit" class="interlanguage-link-target"><span>Taclḥit</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Equation" title="Equation - Simple English" lang="en-simple" hreflang="en-simple" data-title="Equation" 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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Rovnica_(matematika)" title="Rovnica (matematika) - Slofaceg" lang="sk" hreflang="sk" data-title="Rovnica (matematika)" data-language-autonym="Slovenčina" data-language-local-name="Slofaceg" 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/Ena%C4%8Dba" title="Enačba - Slofeneg" lang="sl" hreflang="sl" data-title="Enačba" data-language-autonym="Slovenščina" data-language-local-name="Slofeneg" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Enzane" title="Enzane - Shona" lang="sn" hreflang="sn" data-title="Enzane" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Ekuacioni" title="Ekuacioni - Albaneg" lang="sq" hreflang="sq" data-title="Ekuacioni" data-language-autonym="Shqip" data-language-local-name="Albaneg" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%88%D0%B5%D0%B4%D0%BD%D0%B0%D1%87%D0%B8%D0%BD%D0%B0" title="Једначина - Serbeg" lang="sr" hreflang="sr" data-title="Једначина" data-language-autonym="Српски / srpski" data-language-local-name="Serbeg" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ekvation" title="Ekvation - Swedeg" lang="sv" hreflang="sv" data-title="Ekvation" data-language-autonym="Svenska" data-language-local-name="Swedeg" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Mlinganyo" title="Mlinganyo - Swahili" lang="sw" hreflang="sw" data-title="Mlinganyo" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AE%AE%E0%AE%A9%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81" title="சமன்பாடு - Tamileg" lang="ta" hreflang="ta" data-title="சமன்பாடு" data-language-autonym="தமிழ்" data-language-local-name="Tamileg" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%9C%D1%83%D0%BE%D0%B4%D0%B8%D0%BB%D0%B0" title="Муодила - Tajiceg" lang="tg" hreflang="tg" data-title="Муодила" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajiceg" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AA%E0%B8%A1%E0%B8%81%E0%B8%B2%E0%B8%A3" title="สมการ - Thai" lang="th" hreflang="th" data-title="สมการ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/De%C5%88leme" title="Deňleme - Tyrcmeneg" lang="tk" hreflang="tk" data-title="Deňleme" data-language-autonym="Türkmençe" data-language-local-name="Tyrcmeneg" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Ekwasyon" title="Ekwasyon - Tagalog" lang="tl" hreflang="tl" data-title="Ekwasyon" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Denklem" title="Denklem - Tyrceg" lang="tr" hreflang="tr" data-title="Denklem" data-language-autonym="Türkçe" data-language-local-name="Tyrceg" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A2%D0%B8%D0%B3%D0%B5%D0%B7%D0%BB%D3%99%D0%BC%D3%99" title="Тигезләмә - Tatareg" lang="tt" hreflang="tt" data-title="Тигезләмә" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatareg" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D1%96%D0%B2%D0%BD%D1%8F%D0%BD%D0%BD%D1%8F" title="Рівняння - Wcreineg" lang="uk" hreflang="uk" data-title="Рівняння" data-language-autonym="Українська" data-language-local-name="Wcreineg" 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/%D9%85%D8%B3%D8%A7%D9%88%D8%A7%D8%AA" title="مساوات - Wrdw" lang="ur" hreflang="ur" data-title="مساوات" data-language-autonym="اردو" data-language-local-name="Wrdw" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz badge-Q17437798 badge-goodarticle mw-list-item" title="erthygl dda"><a href="https://uz.wikipedia.org/wiki/Tenglama" title="Tenglama - Wsbeceg" lang="uz" hreflang="uz" data-title="Tenglama" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Wsbeceg" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C6%B0%C6%A1ng_tr%C3%ACnh" title="Phương trình - Fietnameg" lang="vi" hreflang="vi" data-title="Phương trình" data-language-autonym="Tiếng Việt" data-language-local-name="Fietnameg" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Vergelykinge_(wiskunde)" title="Vergelykinge (wiskunde) - Fflemeg Gorllewinol" lang="vls" hreflang="vls" data-title="Vergelykinge (wiskunde)" data-language-autonym="West-Vlams" data-language-local-name="Fflemeg Gorllewinol" class="interlanguage-link-target"><span>West-Vlams</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Ekwasyon" title="Ekwasyon - Winarayeg" lang="war" hreflang="war" data-title="Ekwasyon" data-language-autonym="Winaray" data-language-local-name="Winarayeg" 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/%E6%96%B9%E7%A8%8B" title="方程 - Wu Tsieineaidd" lang="wuu" hreflang="wuu" data-title="方程" data-language-autonym="吴语" data-language-local-name="Wu Tsieineaidd" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/I-Equation" title="I-Equation - Xhosa" lang="xh" hreflang="xh" data-title="I-Equation" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%92%D7%9C%D7%99%D7%99%D7%9B%D7%95%D7%A0%D7%92" title="גלייכונג - Iddew-Almaeneg" lang="yi" hreflang="yi" data-title="גלייכונג" data-language-autonym="ייִדיש" data-language-local-name="Iddew-Almaeneg" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo mw-list-item"><a href="https://yo.wikipedia.org/wiki/%C3%8C%E1%B9%A3ed%E1%BB%8D%CC%81gba" title="Ìṣedọ́gba - Iorwba" lang="yo" hreflang="yo" data-title="Ìṣedọ́gba" data-language-autonym="Yorùbá" data-language-local-name="Iorwba" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%96%B9%E7%A8%8B" title="方程 - Tsieinëeg" lang="zh" hreflang="zh" data-title="方程" data-language-autonym="中文" data-language-local-name="Tsieinëeg" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%96%B9%E7%A8%8B" title="方程 - Literary Chinese" lang="lzh" hreflang="lzh" data-title="方程" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Hong-t%C3%AAng-sek" title="Hong-têng-sek - Minnan" lang="nan" hreflang="nan" data-title="Hong-têng-sek" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%96%B9%E7%A8%8B%E5%BC%8F" title="方程式 - Cantoneeg" lang="yue" hreflang="yue" data-title="方程式" data-language-autonym="粵語" data-language-local-name="Cantoneeg" class="interlanguage-link-target"><span>粵語</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/Q11345#sitelinks-wikipedia" title="Golygu dolenni rhyngwici" class="wbc-editpage">Golygu dolenni</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="Parthau"> <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/Hafaliad" title="Gweld y dudalen bwnc [c]" accesskey="c"><span>Erthygl</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Sgwrs:Hafaliad&action=edit&redlink=1" rel="discussion" class="new" title="Sgwrsio am y dudalen (dim tudalen ar gael) [t]" accesskey="t"><span>Sgwrs</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="Newid amrywiad iaith" > <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">Cymraeg</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="Golygon"> <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/Hafaliad"><span>Darllen</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Hafaliad&veaction=edit" title="Golygu'r dudalen hon [v]" accesskey="v"><span>Golygu</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Hafaliad&action=edit" title="Golygu cod ffynhonnell y dudalen hon [e]" accesskey="e"><span>Golygu cod</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Hafaliad&action=history" title="Fersiynau cynt o'r dudalen hon. [h]" accesskey="h"><span>Gweld hanes</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Offer tudalen"> <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="Blwch offer" > <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">Blwch offer</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 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vector-more-collapsible-item mw-list-item"><a href="/wiki/Hafaliad"><span>Darllen</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Hafaliad&veaction=edit" title="Golygu'r dudalen hon [v]" accesskey="v"><span>Golygu</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Hafaliad&action=edit" title="Golygu cod ffynhonnell y dudalen hon [e]" accesskey="e"><span>Golygu cod</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Hafaliad&action=history"><span>Gweld hanes</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Cyffredinol </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Arbennig:WhatLinksHere/Hafaliad" 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<button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">symud i'r bar ochr</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">cuddio</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">Oddi ar Wicipedia</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="cy" dir="ltr"><table class="infobox" style="width:22em"><caption>Hafaliad</caption><tbody><tr><td colspan="2" style="text-align:center"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/Delwedd:First_Equation_Ever.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/First_Equation_Ever.png/220px-First_Equation_Ever.png" decoding="async" width="220" height="23" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/First_Equation_Ever.png/330px-First_Equation_Ever.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/First_Equation_Ever.png/440px-First_Equation_Ever.png 2x" data-file-width="784" data-file-height="83" /></a></span><div>Carreg glo pob hafaliad yw'r arwydd <b>=</b>, ac fe'i defnyddiwyd am y tro cyntaf gan y Cymro Robert Recorde yn yr hafaliad yma, sy'n mynegi 14<i>x</i> + 15 = 71, yn ein nodiant ni heddiw. Allan o'i gyfrol <i>The Whetstone of Witte</i> (1557).</div></td></tr><tr><th scope="row">Math</th><td><a href="/wiki/Fformiwla" title="Fformiwla">fformiwla</a> <span class="penicon autoconfirmed-show"><span class="mw-valign-text-top" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11345?uselang=cy#P279" title="Edit this on Wikidata"><img alt="Edit this on Wikidata" src="//upload.wikimedia.org/wikipedia/commons/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/commons/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/commons/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></span></td></tr><tr><th scope="row">Y gwrthwyneb</th><td>inequation <span class="penicon autoconfirmed-show"><span class="mw-valign-text-top" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11345?uselang=cy#P461" title="Edit this on Wikidata"><img alt="Edit this on Wikidata" src="//upload.wikimedia.org/wikipedia/commons/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/commons/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/commons/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></span></td></tr><tr><th scope="row">Yn cynnwys</th><td><a href="/wiki/Hafalnod" title="Hafalnod">hafalnod</a> <span class="penicon autoconfirmed-show"><span class="mw-valign-text-top" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q11345?uselang=cy#P527" title="Edit this on Wikidata"><img alt="Edit this on Wikidata" src="//upload.wikimedia.org/wikipedia/commons/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/commons/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/commons/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></span></td></tr><tr><td colspan="2" style="text-align:center"><span typeof="mw:File"><span title="Tudalen Comin"><img alt="Tudalen Comin" src="//upload.wikimedia.org/wikipedia/commons/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/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> <a href="https://commons.wikimedia.org/wiki/Category:Equations" class="extiw" title="commons:Category:Equations">Ffeiliau perthnasol ar Gomin Wicimedia</a></td></tr></tbody></table> <p>Gosodiad <a href="/wiki/Mathemateg" title="Mathemateg">mathemategol</a> yw <b>hafaliad</b> (Saesneg: <i>equation</i>), sy'n cynnwys un neu ragor o <a href="/wiki/Newidyn" title="Newidyn">newidynnau</a>. Mae'n ddull o fynegi fod dau wrthrych mathemategol (rhifau, fel arfer) yn union yr un peth. Mynegir hyn yn symbolaidd â'r <b>hafalnod</b>, <b>=</b> , a ddefnyddiwyd yn gyntaf gan y mathemategwr o Gymro, <a href="/wiki/Robert_Recorde" title="Robert Recorde">Robert Recorde</a> (tua 1510 – 1558). Dyma rai enghreifftiau o hafaliadau: </p> <dl><dd>2 + 3 = 5, neu</dd> <dd><i>x</i> − <i>x</i> = 0, neu</dd> <dd><i> x = y </i>, neu</dd> <dd><i>x</i> + 1 = 2.</dd></dl> <p><a href="/wiki/Unfathiannau" class="mw-redirect" title="Unfathiannau">Unfathiannau</a> yw'r cyntaf a'r ail: maent yn wir, pa bynnag werth a gymer y newidynnau ynddynt. Lle nad yw hafaliad yn unfathiant, fe all y gosodiad fod yn wir neu'n anwir yn dibynnu ar werthoedd y newidynnau ynddo. Fe gelwir gwerthoedd o'r newidynnau sy'n peri i'r gosodiad fod yn wir yn <b>wreiddiau</b> (neu datrysiadau) yr hafaliad. Dywedir eu bod yn <b>bodlonni</b> yr hafaliad. Yn y drydedd enghraifft uchod, mae nifer anfeidrol o ddatrysiadau, <i>x = 1 , y = 1</i> er enghraifft. Yn y bedwaredd enghraifft, dim ond un datrysiad, <i>x = 1</i> sy'n bodoli. Dywedir ei fod yn wraidd unigryw. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Priodweddau_elfennol">Priodweddau elfennol</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hafaliad&veaction=edit&section=1" title="Golygu'r adran: Priodweddau elfennol" class="mw-editsection-visualeditor"><span>golygu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hafaliad&action=edit&section=1" title="Edit section's source code: Priodweddau elfennol"><span>golygu cod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mewn <a href="/wiki/Algebra" title="Algebra">algebra</a> elfenol, os yw hafaliad yn wir, fe ellir deillio hafaliad gwir arall ohono wrth wneud y canlynol: </p> <ol><li><a href="/wiki/Adio" title="Adio">Adio</a> rhif i ddwy ochr yr hafaliad.</li> <li><a href="/wiki/Tynnu" title="Tynnu">Tynnu</a> rhif o ddwy ochr yr hafaliad.</li> <li><a href="/wiki/Lluosi" title="Lluosi">Lluosi</a> dwy ochr yr hafaliad â'r un rhif.</li> <li><a href="/wiki/Rhannu" class="mw-redirect" title="Rhannu">Rhannu</a> dwy ochr yr hafaliad ag unrhyw rhif <i>an-sero</i>.</li> <li>Yn gyffredinol, gellir gymhwyso <a href="/wiki/Ffwythiant" title="Ffwythiant">ffwythiant</a> i'r ddwy ochr.</li></ol> <p>Mae hafaledd yn enghraifft o <a href="/w/index.php?title=Perthynas_unfathiant&action=edit&redlink=1" class="new" title="Perthynas unfathiant (dim tudalen ar gael)">berthynas unfathiant</a>. </p><p>Mae datrys hafaliad sy'n cynnwys <a href="/wiki/Newidyn" title="Newidyn">newidynnau</a> yn cynnwys penderfynu pa werthoedd o'r newidynnau sy'n gwneud y cydraddoldeb yn gywir. Gelwir y newidynnau y mae'n rhaid datrys yr hafaliad ar eu cyfer hefyd yn anhysbysion (<i>unknowns</i>), a gelwir gwerthoedd yr anhysbysion sy'n bodloni'r cydraddoldeb yn ddatrysiadau yr hafaliad. Mae dau fath o hafaliad: <a href="/wiki/Unfathiant" title="Unfathiant">unfathiant (<i>identities</i>)</a> a hafaliadau amodol. Mae unfathiant yn wir am holl werthoedd y newidynnau; mae hafaliad amodol, ar y llaw arall, yn wir am werthoedd penodol y newidynnau yn unig.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Ysgrifennir hafaliad fel dau <a href="/wiki/Mynegiad_(mathemateg)" title="Mynegiad (mathemateg)">ymadrodd</a>, wedi'u cysylltu gan <a href="/wiki/Hafalnod" title="Hafalnod">hafalnod</a> ("="), symbol a ddyfeiswyd yn 1557 gan y Cymro <a href="/wiki/Robert_Recorde" title="Robert Recorde">Robert Recorde</a>.<sup id="cite_ref-:1_2-0" class="reference"><a href="#cite_note-:1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Gelwir yr ymadroddion ar ddwy ochr yr hafalnod yn "ochr chwith" ac "ochr dde" yr hafaliad. Yn aml iawn tybir bod ochr dde hafaliad yn sero. Gan dybio nad yw hyn yn lleihau'r cyffredinolrwydd, oherwydd gellir gwireddu hyn trwy dynnu'r ochr dde o'r ddwy ochr. </p><p>Y math mwyaf cyffredin o hafaliad yw hafaliad polynomial (a elwir yn gyffredin hefyd yn <i>hafaliad algebraidd</i>) lle mae'r ddwy ochr yn <a href="/wiki/Polynomial" title="Polynomial">polynomialau</a>. Mae ochrau hafaliad polynomial yn cynnwys un neu fwy o <a href="/wiki/Adio" title="Adio">dermau</a>. Er enghraifft, yr hafaliad </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 Ax^{2}+Bx+C-y=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>B</mi> <mi>x</mi> <mo>+</mo> <mi>C</mi> <mo>−</mo> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax^{2}+Bx+C-y=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48f35a43975baf07af45c8768074725fdcddb7aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.925ex; height:3.009ex;" alt="{\displaystyle Ax^{2}+Bx+C-y=0}"></span></dd></dl> <p>mae ganddo ochr chwith <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax^{2}+Bx+C-y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>B</mi> <mi>x</mi> <mo>+</mo> <mi>C</mi> <mo>−</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax^{2}+Bx+C-y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dba4508f349e30d6fe1032237df53d03b7d5953e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.664ex; height:3.009ex;" alt="{\displaystyle Ax^{2}+Bx+C-y}"></span>, sydd â phedwar 'term' ac ochr dde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>, sy'n cynnwys un term yn unig. Mae enwau'r <a href="/wiki/Newidyn" title="Newidyn">newidynnau</a> yn awgrymu bod <span class="texhtml"><i>x</i></span> ac <span class="texhtml"><i>y</i></span> yn anhysbys, a bod <span class="texhtml"><i>A</i></span>, <span class="texhtml"><i>B</i></span>, ac <span class="texhtml"><i>C</i></span> yn <a href="/w/index.php?title=Paramedr&action=edit&redlink=1" class="new" title="Paramedr (dim tudalen ar gael)">baramedrau</a>, ond fel arfer mae hyn yn cael ei bennu gan y cyd-destun. </p><p>Mae hafaliad yn cyfateb i glorian, lle rhoddir pwysau arni. Pan roddir pwysau cyfartal o rywbeth (ee <a href="/wiki/Grawn" title="Grawn">grawn</a>) yn y ddwy badell, mae'r ddau bwysau yn achosi i'r raddfa fod mewn 'cydbwysedd' a dywedir eu bod yn gyfartal. Os tynnir swm o rawn o un badell o'r glorian, yna mae'n rhaid tynnu swm cyfartal o rawn o'r badell arall i gadw cydbwysedd. Yn fwy cyffredinol, mae hafaliad yn parhau i fod mewn cydbwysedd os cyflawnir yr un gweithrediad ar y ddwy ochr. </p><p>Mewn <a href="/wiki/Geometreg_ddadansoddol" title="Geometreg ddadansoddol">geometreg Cartesaidd</a> (neu 'ddadansoddol'), defnyddir hafaliadau i ddisgrifio ffigurau geometrig. Gan fod gan yr hafaliadau sy'n cael eu hystyried, fel hafaliadau ymhlyg (<i>Implicit function</i>) neu hafaliadau parametrig, lawer o atebion, mae'r amcan bellach yn wahanol: yn lle rhoi'r atebion yn benodol neu eu cyfrif, sy'n amhosibl, defnyddir yr hafaliadau i astudio priodweddau ffigurau. Dyma'r syniad cychwynnol yr hyn a elwir yn <a href="/wiki/Geometreg_algebraidd" title="Geometreg algebraidd">geometreg algebraidd</a>, maes pwysig o fathemateg. </p><p><a href="/wiki/Algebra" title="Algebra">Mae Algebra</a>'n astudio dau brif deulu o hafaliadau: hafaliadau polynomial a <a href="/wiki/Hafaliad_llinol" title="Hafaliad llinol">hafaliadau llinol</a>. Pan nad oes ond un newidyn, mae gan hafaliadau polynomial y ffurf <i>P</i>(<i>x</i>) = 0, lle mae <i>P</i> yn <a href="/wiki/Polynomial" title="Polynomial">polynomial</a>, ac mae gan hafaliadau llinol y ffurf <i>ax</i> + <i>b</i> = 0, lle <i>mae</i> a a <i>b</i> yn <a href="/w/index.php?title=Paramedr&action=edit&redlink=1" class="new" title="Paramedr (dim tudalen ar gael)">baramedrau</a>. I ddatrys hafaliadau gan y naill deulu neu'r llall, defnyddir technegau algorithmig neu geometrig sy'n tarddu o <a href="/wiki/Algebra_llinol" title="Algebra llinol">algebra llinol</a> neu <a href="/wiki/Dadansoddiad_mathemategol" title="Dadansoddiad mathemategol">ddadansoddiad mathemategol</a>. Mae Algebra hefyd yn astudiaeth o hafaliadau Diophantine lle mae'r cyfernodau a'r datrusiadau yn <a href="/wiki/Cyfanrif" title="Cyfanrif">gyfanrifau</a>. Mae'r technegau a ddefnyddir yn wahanol ac yn dod o <a href="/wiki/Damcaniaeth_rhifau" title="Damcaniaeth rhifau">theori rhif</a>. Mae'r hafaliadau hyn yn anodd yn gyffredinol; mae person yn aml yn chwilio dim ond i ddarganfod bodolaeth neu absenoldeb datrysiad, ac, os ydynt yn bodoli, i gyfrif nifer yr atebion. </p><p>Mae <a href="/wiki/Hafaliad_differol" title="Hafaliad differol">hafaliadau</a> <a href="/wiki/Hafaliad_differol" title="Hafaliad differol">differol</a> (neu 'wahaniaethol') yn hafaliadau sy'n cynnwys un neu fwy o swyddogaethau a'u deilliadau. Fe'u <i>datrysir</i> trwy ddod o hyd i fynegiad ar gyfer y <a href="/wiki/Ffwythiant" title="Ffwythiant">ffwythiant</a> nad yw'n cynnwys <a href="/wiki/Deilliadau" class="mw-redirect" title="Deilliadau">deilliadau</a>. Defnyddir hafaliadau gwahaniaethol i fodelu prosesau sy'n cynnwys cyfraddau newid y newidyn, ac fe'u defnyddir mewn meysydd fel ffiseg, cemeg, bioleg ac economeg. </p> <div class="mw-heading mw-heading2"><h2 id="Cyflwyniad">Cyflwyniad</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hafaliad&veaction=edit&section=2" title="Golygu'r adran: Cyflwyniad" class="mw-editsection-visualeditor"><span>golygu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hafaliad&action=edit&section=2" title="Edit section's source code: Cyflwyniad"><span>golygu cod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Darlun_analog">Darlun analog</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hafaliad&veaction=edit&section=3" title="Golygu'r adran: Darlun analog" class="mw-editsection-visualeditor"><span>golygu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hafaliad&action=edit&section=3" title="Edit section's source code: Darlun analog"><span>golygu cod</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Delwedd:Equation_illustration_colour.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Equation_illustration_colour.svg/220px-Equation_illustration_colour.svg.png" decoding="async" width="220" height="230" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Equation_illustration_colour.svg/330px-Equation_illustration_colour.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Equation_illustration_colour.svg/440px-Equation_illustration_colour.svg.png 2x" data-file-width="278" data-file-height="291" /></a><figcaption> Darlun o hafaliad syml; <i>Mae x</i>, <i>y</i>, <i>z</i> yn rhifau real, sy'n cyfateb i bwysau.</figcaption></figure> <p>Mae hafaliad yn cyfateb i glorian neu si-so. </p><p>Yn y llun, mae <i>x</i>, <i>y</i> a <i>z</i> i gyd yn feintiau gwahanol (yn yr achos hwn <a href="/wiki/Rhif_real" title="Rhif real">rhifau real</a> ) a gynrychiolir fel pwysau crwn, ac mae pwysau gwahanol ar <i>bob un o x</i>, <i>y</i> <i>, a z.</i> Mae adio yn cyfateb i ychwanegu pwysau, tra bod tynnu yn cyfateb i dynnu pwysau o'r hyn sydd yno eisoes. Pan fydd cydraddoldeb, mae cyfanswm y pwysau ar bob ochr yr un peth. </p> <div class="mw-heading mw-heading3"><h3 id="Paramedrau_ac_anhysbysion">Paramedrau ac anhysbysion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hafaliad&veaction=edit&section=4" title="Golygu'r adran: Paramedrau ac anhysbysion" class="mw-editsection-visualeditor"><span>golygu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hafaliad&action=edit&section=4" title="Edit section's source code: Paramedrau ac anhysbysion"><span>golygu cod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mae hafaliadau yn aml yn cynnwys termau heblaw'r anhysbysion. Fel rheol, gelwir y termau eraill hyn, y tybir eu bod yn <i>hysbys</i> yn <i>gysonion</i>, <i>cyfernodau</i> neu <i>baramedrau</i>. </p><p>Enghraifft o hafaliad sy'n cynnwys <i>x</i> ac <i>y</i> fel anhysbysion a'r paramedr <i>R</i> yw </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}+y^{2}=R^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=R^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe1f27078e621122772fdeb967fe85a71be259b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.003ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=R^{2}.}"></span></dd></dl> <p>Pan <i>ddewisir R</i> i fod â gwerth 2 ( <i>R</i> = 2), byddai'r hafaliad hwn yn cael ei gydnabod mewn <a href="/wiki/System_gyfesurynnol_Gartesaidd" title="System gyfesurynnol Gartesaidd">cyfesurynnau Cartesaidd</a> fel yr hafaliad ar gyfer y cylch radiws o 2 o amgylch y tardd. Felly, yr hafaliad gydag <i>R</i> amhenodol yw'r hafaliad cyffredinol ar gyfer y cylch. </p><p>Fel arfer, dynodir yr anhysbysion gan lythrennau ar ddiwedd yr wyddor, <i>x</i>, <i>y</i>, <i>z</i>, <i>w</i>, ..., tra bod cyfernodau (paramedrau) yn cael eu dynodi gan lythrennau ar ddechrau'r wyddor, <i>a</i>, <i>b</i>, <i>c</i>, <i>d,.</i> . . . Er enghraifft, yr <a href="/wiki/Hafaliad_cwadratig" title="Hafaliad cwadratig">hafaliad cwadratig</a> cyffredinol fel arfer yw <i>ax</i><sup>2</sup> + <i>bx</i> + <i>c</i> = 0. </p><p>Gelwir y broses o ddod o hyd i'r atebion, neu, yn achos y paramedrau, mynegi'r anhysbys yn nhermau'r paramedrau, yn 'ddatrys yr hafaliad'. Gelwir mynegiadau o'r fath o'r atebion o ran y paramedrau hefyd yn <i>atebion</i>. </p><p>Mae system o hafaliadau yn set o <i>hafaliadau cydamserol (simultaneous equations)</i>, fel arfer mewn sawl anhysbys y ceisir yr atebion cyffredin ar eu cyfer. Felly, mae'r <i>ateb i'r system</i> yn set o werthoedd ar gyfer pob un o'r anhysbysion, sydd gyda'i gilydd yn ffurfio ateb i bob hafaliad yn y system. Er enghraifft, mae gan y system </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}3x+5y&=2\\5x+8y&=3\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>x</mi> <mo>+</mo> <mn>5</mn> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>5</mn> <mi>x</mi> <mo>+</mo> <mn>8</mn> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}3x+5y&=2\\5x+8y&=3\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64d70f591a29304ca95853d2b2ddd15fe3b67b94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.663ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}3x+5y&=2\\5x+8y&=3\end{aligned}}}"></span></dd></dl> <p>y datrysiad unigryw <i>x</i> = −1, <i>y</i> = 1. </p> <div class="mw-heading mw-heading3"><h3 id="Unfathiant">Unfathiant</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hafaliad&veaction=edit&section=5" title="Golygu'r adran: Unfathiant" class="mw-editsection-visualeditor"><span>golygu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hafaliad&action=edit&section=5" title="Edit section's source code: Unfathiant"><span>golygu cod</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Delwedd:Trig_functions_on_unit_circle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Trig_functions_on_unit_circle.svg/220px-Trig_functions_on_unit_circle.svg.png" decoding="async" width="220" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Trig_functions_on_unit_circle.svg/330px-Trig_functions_on_unit_circle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/Trig_functions_on_unit_circle.svg/440px-Trig_functions_on_unit_circle.svg.png 2x" data-file-width="608" data-file-height="560" /></a><figcaption>Prawf gweledol o gyfathiant Pythagoraidd. Ar gyfer unrhyw <a href="/wiki/Ongl" title="Ongl">ongl</a> θ, mae'r <a href="/wiki/Pwynt_(geometreg)" title="Pwynt (geometreg)">pwynt</a> (cos(θ),sin(θ)) yn gorwedd ar y cylch, sy'n bodloni yr <a class="mw-selflink selflink">hafaliad</a> <i>x</i><sup>2</sup>+<i>y</i><sup>2</sup>=1. Felly, cos<sup>2</sup>(θ)+sin<sup>2</sup>(θ)=1.</figcaption></figure> <p>Mewn <a href="/wiki/Mathemateg" title="Mathemateg">mathemateg</a> unfathiant (Saesneg: <i>identity</i>) yw perthynas yr <a href="/wiki/Hafaledd" title="Hafaledd">hafaledd</a> <i>A</i> = <i>B</i>, fel bod <i>A</i> a <i>B</i> yn cynnwys rhai <a href="/wiki/Newidyn" title="Newidyn">newidynnau</a> a lle mae <i>A</i> a <i>B</i> yn rhoi'r un gwerthoedd a'i gilydd, ni waeth be fo'r gwerthoedd (rhifau, fel arfer) a gaiff eu cyfnewid am newidynnau. Mewn geiriau eraill, mae <i>A</i> = <i>B</i> yn unfathiant os yw <i>A</i> a <i>B</i> yn diffinio yr un <a href="/wiki/Ffwythiant" title="Ffwythiant">ffwythiannau</a>. Golyga hyn fod yr 'unfathiant' yn 'hafaledd' (<i>equality</i>) rhwng ffwythiannau a ddiffiniwyd yn wahanol. Er enghraifft, mae (<i>a</i> + <i>b</i>)<sup>2</sup>  =  <i>a</i><sup>2</sup> + 2<i>ab</i> + <i>b</i><sup>2</sup> a <span class="nowrap">cos<sup>2</sup>(<i>x</i>) + sin<sup>2</sup>(<i>x</i>) = 1</span> yn unfathiannau. </p><p>Caiff unfathiannau eu <a href="/wiki/Nodiant_mathemategol" title="Nodiant mathemategol">dynodi</a> gan y symbol <big><span class="texhtml">≡</span></big> (bariau triphlyg ), yn hytrach na <span class="texhtml">=</span>, sef yr <a class="mw-selflink selflink">hafaliad</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Mewn geiriau eraill: enghraifft o unfathiant yw'r gwahaniaeth rhwng dau sgwâr: </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}-y^{2}=(x+y)(x-y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}-y^{2}=(x+y)(x-y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6364fdb9e9d31860302d0d4dd231cc4f06e992c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.807ex; height:3.176ex;" alt="{\displaystyle x^{2}-y^{2}=(x+y)(x-y)}"></span></dd></dl> <p>sy'n wir am bob <i>x</i> ac <i>y</i>. </p><p>Mae <a href="/wiki/Trigonometreg" title="Trigonometreg">trigonometreg</a> yn faes lle mae llawer o unfathianau'n bodoli; mae'r rhain yn ddefnyddiol wrth drin neu ddatrys <a href="/wiki/Rhestr_unfathiannau_trigonometrig" title="Rhestr unfathiannau trigonometrig">hafaliadau trigonometrig</a>. Dau o lawer sy'n cynnwys y swyddogaethau sin a <a href="/wiki/Ffwythiannau_trigonometrig" title="Ffwythiannau trigonometrig">chosin</a> yw: </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 \sin ^{2}(\theta )+\cos ^{2}(\theta )=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}(\theta )+\cos ^{2}(\theta )=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47f3203bca6dc55c36d94ee525c44dac9e1716f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.976ex; height:3.176ex;" alt="{\displaystyle \sin ^{2}(\theta )+\cos ^{2}(\theta )=1}"></span></dd></dl> <p>lle mae <i>θ</i> wedi'i gyfyngu i rhwng 0 a 45 gradd, gall un ddefnyddio'r hunaniaeth uchod i'r cynnyrch roi: </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 \sin(2\theta )=2\sin(\theta )\cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(2\theta )=2\sin(\theta )\cos(\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49b9120d4e69a660935b978d66f352fd2e645199" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.719ex; height:2.843ex;" alt="{\displaystyle \sin(2\theta )=2\sin(\theta )\cos(\theta )}"></span></dd></dl> <p>sydd ill dau'n wir am holl werthoedd <i>θ</i>. </p><p>Er enghraifft, i ddatrys gwerth <i>θ</i> sy'n bodloni'r hafaliad: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\sin(\theta )\cos(\theta )=1\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\sin(\theta )\cos(\theta )=1\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef5afc15255c1fbbb969663c81461fde3c67f197" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.998ex; height:2.843ex;" alt="{\displaystyle 3\sin(\theta )\cos(\theta )=1\,,}"></span></dd></dl> <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 {\frac {3}{2}}\sin(2\theta )=1\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3}{2}}\sin(2\theta )=1\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d40af5eefb61021cfcf56a14ecc5407d8f2417e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.598ex; height:5.176ex;" alt="{\displaystyle {\frac {3}{2}}\sin(2\theta )=1\,,}"></span></dd></dl> <p>sy'n ildio'r datrysiad canlynol ar gyfer <i>θ:</i> </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 \theta ={\frac {1}{2}}\arcsin \left({\frac {2}{3}}\right)\approx 20.9^{\circ }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <msup> <mn>20.9</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ={\frac {1}{2}}\arcsin \left({\frac {2}{3}}\right)\approx 20.9^{\circ }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d051c1ae231480e303ae4e8291dfe51173147613" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.89ex; height:6.176ex;" alt="{\displaystyle \theta ={\frac {1}{2}}\arcsin \left({\frac {2}{3}}\right)\approx 20.9^{\circ }.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Priodweddau">Priodweddau</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hafaliad&veaction=edit&section=6" title="Golygu'r adran: Priodweddau" class="mw-editsection-visualeditor"><span>golygu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hafaliad&action=edit&section=6" title="Edit section's source code: Priodweddau"><span>golygu cod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mae dwy hafaliad neu ddwy system o hafaliadau yn <i>gyfwerth</i>, os oes ganddyn nhw'r un set o atebion. Mae'r gweithrediadau canlynol yn trawsnewid hafaliad neu system o hafaliadau yn un gyfwerth - ar yr amod bod y gweithrediadau yn ystyrlon ar gyfer yr ymadroddion y maent yn berthnasol iddynt: </p> <ul><li><a href="/wiki/Adio" title="Adio">Adio</a> neu <a href="/wiki/Tynnu" title="Tynnu">dynnu'r</a> un maint i (neu o) ddwy ochr hafaliad. Mae hyn yn dangos bod pob hafaliad yn cyfateb i hafaliad lle mae'r ochr dde yn sero.</li> <li><a href="/wiki/Lluosi" title="Lluosi">Lluosi</a> neu <a href="/wiki/Rhannu_(mathemateg)" title="Rhannu (mathemateg)">rannu</a> dwy ochr hafaliad â maint nad yw'n sero.</li> <li>Cymhwyso <a href="/wiki/Unfathiant" title="Unfathiant">unfathiant</a> i drawsnewid un ochr i'r hafaliad. Er enghraifft, ehangu lluoswm neu ffactoreiddio swm.</li> <li>Ar gyfer system: ychwanegu ochr gyfatebol hafaliad arall i ddwy ochr hafaliad, wedi'i luosi â'r un maint.</li></ul> <p>Er enghraifft, mae </p> <div class="mw-heading mw-heading2"><h2 id="Algebra">Algebra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hafaliad&veaction=edit&section=7" title="Golygu'r adran: Algebra" class="mw-editsection-visualeditor"><span>golygu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hafaliad&action=edit&section=7" title="Edit section's source code: Algebra"><span>golygu cod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Hafaliadau_polynomial">Hafaliadau polynomial</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hafaliad&veaction=edit&section=8" title="Golygu'r adran: Hafaliadau polynomial" class="mw-editsection-visualeditor"><span>golygu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hafaliad&action=edit&section=8" title="Edit section's source code: Hafaliadau polynomial"><span>golygu cod</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Delwedd:Polynomialdeg2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Polynomialdeg2.svg/220px-Polynomialdeg2.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Polynomialdeg2.svg/330px-Polynomialdeg2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Polynomialdeg2.svg/440px-Polynomialdeg2.svg.png 2x" data-file-width="320" data-file-height="320" /></a><figcaption> <i>Datrysiadau</i> –1 a 2 yr <i>hafaliad polynomial</i> <span class="nowrap"><i>x</i><sup>2</sup> – <i>x</i> + 2 = 0</span> yw'r pwyntiau lle mae graff y <a href="/wiki/Polynomial_cwadratig" title="Polynomial cwadratig">swyddogaeth gwadratig</a> <span class="nowrap"><i>y</i> = <i>x</i><sup>2</sup> – <i>x</i> + 2</span> yn torri'r echelin-<i>x</i>.</figcaption></figure> <p>Yn gyffredinol, <i>hafaliad algebraidd</i> neu hafaliad polynomial yw hafaliad o'r ffurf </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 P=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6f743f37b37ce0c2ddc1db0fdca0e577c19f51d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle P=0}"></span>, neu</dd></dl> <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 P=Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2abc7e2c5a78e9e6cb7a2a907279953f9b4a3f52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.682ex; height:2.509ex;" alt="{\displaystyle P=Q}"></span></dd></dl> <p>lle mae <i>P</i> a <i>Q</i> yn <a href="/wiki/Polynomial" title="Polynomial">polynomialau</a> â <a href="/wiki/Cyfernod" title="Cyfernod">chyfernodau</a> mewn rhai maes (ee <a href="/wiki/Rhif_cymarebol" title="Rhif cymarebol">rhifau rhesymegol</a>, <a href="/wiki/Rhif_real" title="Rhif real">rhifau real</a>, <a href="/wiki/Rhif_cymhlyg" title="Rhif cymhlyg">rhifau cymhlyg</a>). Mae hafaliad algebraidd yn <i>un-amrywedd</i> (<i>univariate)</i> os yw'n cynnwys un <a href="/wiki/Newidyn" title="Newidyn">newidyn yn</a> unig. Ar y llaw arall, gall hafaliad polynomial gynnwys sawl newidyn, ac os felly fe'i gelwir yn <i>aml-amrywedd</i> (<i>multivariate;</i> newidynnau lluosog, x, y, z, ac ati). Mae'r term <i>hafaliad polynomial</i> fel arfer yn cael ei ddefnyddio'n hytrach <i>nag hafaliad algebraidd</i>. </p><p>Mae <a href="/w/index.php?title=System_hafaliadau_llinol&action=edit&redlink=1" class="new" title="System hafaliadau llinol (dim tudalen ar gael)">system hafaliadau llinol</a> (neu <i>system linellol</i>) yn gasgliad o <a href="/wiki/Hafaliad_llinol" title="Hafaliad llinol">hafaliadau llinol</a> sy'n cynnwys yr un set o <a href="/wiki/Newidyn" title="Newidyn">newidynnau</a>. Er enghraifft, mae </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{5}-3x+1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{5}-3x+1=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/698e1afb9a1d47e492390b6a5a4612ea0dfff0cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.98ex; height:2.843ex;" alt="{\displaystyle x^{5}-3x+1=0}"></span></dd></dl> <p>yn hafaliad algebraidd (polynomial) yn <i>univariate</i> gyda chyfernodau cyfanrif 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 y^{4}+{\frac {xy}{2}}={\frac {x^{3}}{3}}-xy^{2}+y^{2}-{\frac {1}{7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{4}+{\frac {xy}{2}}={\frac {x^{3}}{3}}-xy^{2}+y^{2}-{\frac {1}{7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5e8a49bf6d100e7a8b65135f6faffd17e470ceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.974ex; height:5.843ex;" alt="{\displaystyle y^{4}+{\frac {xy}{2}}={\frac {x^{3}}{3}}-xy^{2}+y^{2}-{\frac {1}{7}}}"></span></dd></dl> <p>yn hafaliad polynomial aml-amrywedd dros y rhifau rhesymegol. </p> <div class="mw-heading mw-heading3"><h3 id="Systemau_hafaliadau_llinol">Systemau hafaliadau llinol</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hafaliad&veaction=edit&section=9" title="Golygu'r adran: Systemau hafaliadau llinol" class="mw-editsection-visualeditor"><span>golygu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hafaliad&action=edit&section=9" title="Edit section's source code: Systemau hafaliadau llinol"><span>golygu cod</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Delwedd:%E4%B9%9D%E7%AB%A0%E7%AE%97%E8%A1%93.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/%E4%B9%9D%E7%AB%A0%E7%AE%97%E8%A1%93.gif/220px-%E4%B9%9D%E7%AB%A0%E7%AE%97%E8%A1%93.gif" decoding="async" width="220" height="287" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/%E4%B9%9D%E7%AB%A0%E7%AE%97%E8%A1%93.gif/330px-%E4%B9%9D%E7%AB%A0%E7%AE%97%E8%A1%93.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/8/88/%E4%B9%9D%E7%AB%A0%E7%AE%97%E8%A1%93.gif 2x" data-file-width="419" data-file-height="546" /></a><figcaption> Llyfr Tsieineaidd dienw yw'r <i>Naw Pennod ar y Gelf Fathemategol</i> sy'n cynnig dull datrys ar gyfer hafaliadau llinol.</figcaption></figure> <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}3x&&\;+\;&&2y&&\;-\;&&z&&\;=\;&&1&\\2x&&\;-\;&&2y&&\;+\;&&4z&&\;=\;&&-2&\\-x&&\;+\;&&{\tfrac {1}{2}}y&&\;-\;&&z&&\;=\;&&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> <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> <mi>z</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> <mn>2</mn> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>4</mn> <mi>z</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd /> </mtr> <mtr> <mtd> <mo>−</mo> <mi>x</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>+</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>y</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>−</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mi>z</mi> </mtd> <mtd /> <mtd> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> </mtd> <mtd /> <mtd> <mn>0</mn> </mtd> <mtd /> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{7}3x&&\;+\;&&2y&&\;-\;&&z&&\;=\;&&1&\\2x&&\;-\;&&2y&&\;+\;&&4z&&\;=\;&&-2&\\-x&&\;+\;&&{\tfrac {1}{2}}y&&\;-\;&&z&&\;=\;&&0&\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d691839a2b284331b58b0820654d32e101e26a03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:21.219ex; height:9.676ex;" alt="{\displaystyle {\begin{alignedat}{7}3x&&\;+\;&&2y&&\;-\;&&z&&\;=\;&&1&\\2x&&\;-\;&&2y&&\;+\;&&4z&&\;=\;&&-2&\\-x&&\;+\;&&{\tfrac {1}{2}}y&&\;-\;&&z&&\;=\;&&0&\end{alignedat}}}"></span></dd></dl> <p>yn system o dri hafaliad yn y tri newidyn x, y, z . Datrysiad i system linellol yw aseiniad rhifau i'r newidynnau fel bod yr holl hafaliadau'n cael eu bodloni ar yr un pryd. Rhoddir datrysiad i'r system uchod gan </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}{2}x&\,=\,&1\\y&\,=\,&-2\\z&\,=\,&-2\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}x&\,=\,&1\\y&\,=\,&-2\\z&\,=\,&-2\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4460866202afb67e822389a6b11a0b453c89c1c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:8.279ex; height:8.843ex;" alt="{\displaystyle {\begin{alignedat}{2}x&\,=\,&1\\y&\,=\,&-2\\z&\,=\,&-2\end{alignedat}}}"></span></dd></dl> <p>gan ei fod yn gwneud y tri hafaliad yn ddilys. Mae'r gair "<i>system</i>" yn nodi bod yr hafaliadau i'w hystyried ar y cyd, yn hytrach nag yn unigol. </p><p>Mewn mathemateg, theori systemau llinol yw sylfaen a rhan sylfaenol <a href="/wiki/Algebra_llinol" title="Algebra llinol">algebra llinol</a>, pwnc a ddefnyddir yn y rhan fwyaf o fathemateg fodern. Mae <a href="/wiki/Algorithm" title="Algorithm">algorithmau</a> cyfrifiadol ar gyfer dod o hyd i'r atebion yn rhan bwysig o algebra llinol rhifiadol, ac maent yn chwarae rhan amlwg mewn <a href="/wiki/Ffiseg" title="Ffiseg">ffiseg</a>, <a href="/wiki/Peirianneg" title="Peirianneg">peirianneg</a>, <a href="/wiki/Cemeg" title="Cemeg">cemeg</a>, <a href="/wiki/Cyfrifiadureg" title="Cyfrifiadureg">gwyddoniaeth gyfrifiadurol</a>, ac <a href="/wiki/Economeg" title="Economeg">economeg</a>. Gellir <a href="/wiki/Brasamcan" title="Brasamcan">brasamcanu</a> system o hafaliadau aflinol yn aml drwy system llinol, techneg ddefnyddiol wrth wneud model mathemategol neu efelychiad cyfrifiadurol o system gymharol gymhlyg. </p> <div class="mw-heading mw-heading2"><h2 id="Geometreg">Geometreg</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hafaliad&veaction=edit&section=10" title="Golygu'r adran: Geometreg" class="mw-editsection-visualeditor"><span>golygu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hafaliad&action=edit&section=10" title="Edit section's source code: Geometreg"><span>golygu cod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Geometreg_ddadansoddol">Geometreg ddadansoddol</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hafaliad&veaction=edit&section=11" title="Golygu'r adran: Geometreg ddadansoddol" class="mw-editsection-visualeditor"><span>golygu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hafaliad&action=edit&section=11" title="Edit section's source code: Geometreg ddadansoddol"><span>golygu cod</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Delwedd:Coniques_cone.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Coniques_cone.png/220px-Coniques_cone.png" decoding="async" width="220" height="138" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Coniques_cone.png/330px-Coniques_cone.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Coniques_cone.png/440px-Coniques_cone.png 2x" data-file-width="536" data-file-height="335" /></a><figcaption> Rhan <a href="/wiki/Trychiad_conig" title="Trychiad conig">conig</a> yw croestoriad plân a chôn tro.</figcaption></figure> <p>Mewn <a href="/wiki/Geometreg_Euclidaidd" title="Geometreg Euclidaidd">geometreg Ewclidaidd</a>, mae'n bosibl cysylltu set o gyfesurynnau â phob pwynt yn y gofod, er enghraifft gan grid orthogonal. Mae'r dull hwn yn caniatáu i un nodweddu ffigurau geometrig drwy hafaliadau. Gellir mynegi <a href="/wiki/Pl%C3%A2n" class="mw-redirect" title="Plân">plân</a> mewn gofod tri dimensiwn fel set i ateb hafaliad y ffurf <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax+by+cz+d=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo>+</mo> <mi>c</mi> <mi>z</mi> <mo>+</mo> <mi>d</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax+by+cz+d=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/746e9e9c5e6a562485cbc5ed6a9375ec67bad26f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.805ex; height:2.509ex;" alt="{\displaystyle ax+by+cz+d=0}"></span>, lle mae <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}"></span> 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 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/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> yn rhifau real 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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbeca34b28f569a407ef74a955d041df9f360268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.641ex; height:2.009ex;" alt="{\displaystyle x,y,z}"></span> yw'r pethau anhysbys sy'n cyfateb i gyfesurynnau pwynt yn y system a roddir gan y grid orthogonal. Y gwerthoedd <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}"></span> yw cyfesurynnau fector sy'n berpendicwlar i'r plân a ddiffinnir gan yr hafaliad. Gall llinell gael ei mynegi fel croestoriad dwy blân, hynny yw fel set sy'n ateb yr hafaliad llinol sengl gyda gwerthoedd yn <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 \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span> neu fel set datrysiad dau hafaliad llinol â gwerthoedd yn <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 \mathbb {R} ^{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b00b2b4fd27c2cbffa02df568472f77b194a6db9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}.}"></span> </p><p><a href="/wiki/Trychiad_conig" title="Trychiad conig">Adran conig</a> yw croestoriad <a href="/wiki/C%C3%B4n" title="Côn">côn</a> ag hafaliad <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}+y^{2}=z^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}=z^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24defb565dbcba7850e1bfb51176bcf574b4e56b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.682ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}=z^{2}}"></span> a <a href="/wiki/Pl%C3%A2n" class="mw-redirect" title="Plân">phlân</a>. Mewn geiriau eraill, yn y gofod, diffinnir pob conig fel set datrysiad hafaliad plân a hafaliad côn. Mae'r ffurfioldeb hwn yn caniatáu i un bennu lleoliad a phriodweddau ffocysau conig. </p> <div class="mw-heading mw-heading3"><h3 id="Hafaliadau_Cartesaidd">Hafaliadau Cartesaidd</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hafaliad&veaction=edit&section=12" title="Golygu'r adran: Hafaliadau Cartesaidd" class="mw-editsection-visualeditor"><span>golygu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hafaliad&action=edit&section=12" title="Edit section's source code: Hafaliadau Cartesaidd"><span>golygu cod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mae <a href="/wiki/System_gyfesurynnol_Gartesaidd" title="System gyfesurynnol Gartesaidd">system gyfesurynnol Gartesaidd</a> yn <a href="/wiki/System_gyfesurynnau" title="System gyfesurynnau">system</a> <a href="/wiki/System_gyfesurynnau" title="System gyfesurynnau">system o gyfesurynnau</a> sy'n pennu pob <a href="/wiki/Pwynt_(geometreg)" title="Pwynt (geometreg)">pwynt</a> unigryw mewn <a href="/wiki/Pl%C3%A2n_geometraidd" title="Plân geometraidd">plân</a> gan bâr o gyfesurynnau <a href="/wiki/Rhif" title="Rhif">rhifiadol</a><b>,</b> sef y pellteroedd o'r pwynt i ddwy linell sefydlog <a href="/wiki/Perpendicwlar" title="Perpendicwlar">berpendicwlar</a>, sy'n cael eu marcio gan ddefnyddio'r un uned o hyd. </p> <div class="mw-heading mw-heading2"><h2 id="Damcaniaeth_rhif">Damcaniaeth rhif</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hafaliad&veaction=edit&section=13" title="Golygu'r adran: Damcaniaeth rhif" class="mw-editsection-visualeditor"><span>golygu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hafaliad&action=edit&section=13" title="Edit section's source code: Damcaniaeth rhif"><span>golygu cod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Hafaliadau_Diophantine">Hafaliadau Diophantine</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hafaliad&veaction=edit&section=14" title="Golygu'r adran: Hafaliadau Diophantine" class="mw-editsection-visualeditor"><span>golygu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hafaliad&action=edit&section=14" title="Edit section's source code: Hafaliadau Diophantine"><span>golygu cod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mae hafaliad Diophantine yn hafaliad polynomial mewn dau anhysbys neu fwy - lle ceisir dim ond datrysiad cyfanrif (integer solution) ar eu cyfer; mae datrysiad cyfanrif yn ddatrysiad fel bod yr holl bethau anhysbys yn cymryd gwerthoedd cyfanrif). Mae hafaliad Diophantine llinol yn hafaliad rhwng dau swm o fonomial o radd sero neu un. Enghraifft o hafaliad Diophantine llinol yw lle mae a, b, ac c yn gysonion. Mae hafaliad Diophantine esbonyddol yn un y gall esbonwyr termau'r hafaliad fod yn anhysbys ar ei gyfer. </p><p>Mae gan broblemau diophantine lai o hafaliadau na newidynnau anhysbys ac maent yn cynnwys dod o hyd i gyfanrifau sy'n gweithio'n gywir ar gyfer pob hafaliad. Mewn iaith fwy technegol, maent yn diffinio cromlin algebraidd, arwyneb algebraidd, neu wrthrych mwy cyffredinol, ac yn gofyn am y pwyntiau dellt (lattice points) arno. </p><p>Mae'r gair Diophantine yn cyfeirio at fathemategydd Helenistig y 3g, sef Diophantus o Alexandria, a wnaeth astudiaeth o hafaliadau o'r fath ac a oedd yn un o'r mathemategwyr cyntaf i gyflwyno symbolaeth i algebra. Bellach gelwir yr astudiaeth fathemategol o broblemau Diophantine a gychwynnodd Diophantus yn 'ddadansoddiad Diophantine'. </p> <div class="mw-heading mw-heading2"><h2 id="Hafaliadau_differol">Hafaliadau differol</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hafaliad&veaction=edit&section=15" title="Golygu'r adran: Hafaliadau differol" class="mw-editsection-visualeditor"><span>golygu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hafaliad&action=edit&section=15" title="Edit section's source code: Hafaliadau differol"><span>golygu cod</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Delwedd:Attracteur_%C3%A9trange_de_Lorenz.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Attracteur_%C3%A9trange_de_Lorenz.png/220px-Attracteur_%C3%A9trange_de_Lorenz.png" decoding="async" width="220" height="179" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Attracteur_%C3%A9trange_de_Lorenz.png/330px-Attracteur_%C3%A9trange_de_Lorenz.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/Attracteur_%C3%A9trange_de_Lorenz.png/440px-Attracteur_%C3%A9trange_de_Lorenz.png 2x" data-file-width="965" data-file-height="784" /></a><figcaption> Denwr rhyfedd (<i>strange attractor</i>) sy'n codi wrth ddatrys <a href="/wiki/Hafaliad_differol" title="Hafaliad differol">hafaliadau differol, penodol</a></figcaption></figure> <p><a href="/wiki/Hafaliad_differol" title="Hafaliad differol">Hafaliad differol</a> yw <a href="/wiki/Mathemateg" title="Mathemateg">hafaliad mathemategol</a> sy'n cysylltu rhywfaint o <a href="/wiki/Ffwythiant" title="Ffwythiant">ffwythiant</a> â'i <a href="/wiki/Deilliant" title="Deilliant">ddeilliannau</a>. Mewn cymwysiadau, mae'r ffwythiannau fel arfer yn cynrychioli meintiau corfforol, mae'r deilliadau'n cynrychioli eu cyfraddau newid, ac mae'r hafaliad yn diffinio perthynas rhwng y ddau. Oherwydd bod cysylltiadau o'r fath yn hynod gyffredin, mae hafaliadau differol yn chwarae rhan amlwg mewn llawer o ddisgyblaethau gan gynnwys <a href="/wiki/Ffiseg" title="Ffiseg">ffiseg</a>, <a href="/wiki/Peirianneg" title="Peirianneg">peirianneg</a>, <a href="/wiki/Economeg" title="Economeg">economeg</a> a <a href="/wiki/Bywydeg" title="Bywydeg">bioleg</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Geometreg_algebraidd">Geometreg algebraidd</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hafaliad&veaction=edit&section=16" title="Golygu'r adran: Geometreg algebraidd" class="mw-editsection-visualeditor"><span>golygu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hafaliad&action=edit&section=16" title="Edit section's source code: Geometreg algebraidd"><span>golygu cod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mae geometreg algebraidd yn astudiaeth o <a href="/wiki/Sero" class="mw-redirect" title="Sero">seroau</a> ('isradd' neu 'sero'r <a href="/wiki/Ffwythiant" title="Ffwythiant">ffwythiant</a>') <a href="/wiki/Polynomial" title="Polynomial">polynomialau</a> aml-gyfeiriol. Mae geometreg algebraidd modern yn seiliedig ar y defnydd o dechnegau algebra haniaethol, yn bennaf ar gyfer datrys <a href="/wiki/Geometreg" title="Geometreg">problemau geometrig</a> am setiau o seros. Amcanion sylfaenol yr astudiaeth o geometreg algebraidd yw 'amrywiaeth algebraidd'. Amrywiaeth algebraidd, felly, yw prif faes astudiaeth geometreg algebraidd. </p><p>Ymhlith yr amrywiaeth algebraidd a astudir fwyaf aml mae comliniau algebraidd sy'n cynnwys <a href="/wiki/Llinell" title="Llinell">llinellau</a>, <a href="/wiki/Cylch" title="Cylch">cylchoedd</a>, <a href="/wiki/El%C3%ADps" title="Elíps">elipsau</a>, <a href="/wiki/Hyperbola" title="Hyperbola">hyperbolâu</a>, <a href="/wiki/Llinell" title="Llinell">llinellau</a>, <a href="/wiki/Parabola" title="Parabola">parabolâu</a>, cromlinau ciwbic a chromlinau cwartig. Mae geometreg algebraidd yn cymryd lle canolog mewn mathemateg fodern ac mae ganddo lawer o gysylltiadau cysyniadol a meysydd mor amrywiol â <a href="/wiki/Dadansoddiad_cymhleth" class="mw-redirect" title="Dadansoddiad cymhleth">dadansoddiad cymhleth</a>, <a href="/wiki/Topoleg" title="Topoleg">topoleg</a> a <a href="/wiki/Theori_rhif" class="mw-redirect" title="Theori rhif">theori rhif</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Mathau_o_hafaliadau">Mathau o hafaliadau</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hafaliad&veaction=edit&section=17" title="Golygu'r adran: Mathau o hafaliadau" class="mw-editsection-visualeditor"><span>golygu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hafaliad&action=edit&section=17" title="Edit section's source code: Mathau o hafaliadau"><span>golygu cod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gellir dosbarthu hafaliadau yn ôl y mathau o weithrediadau a meintiau dan sylw. Ymhlith y mathau pwysig mae: </p> <ul><li><a href="/w/index.php?title=Hafaliad_algebraidd&action=edit&redlink=1" class="new" title="Hafaliad algebraidd (dim tudalen ar gael)">hafaliad algebraidd</a> neu <a href="/wiki/Polynomial" title="Polynomial">hafaliad polynomial</a>, lle mae'r ddwy ochr yn polynomialau (gweler hefyd system hafaliadau polynomial). Dosberthir y rhain ymhellach yn ôl gradd : <ul><li><a href="/wiki/Hafaliad_llinol" title="Hafaliad llinol">hafaliad llinol</a> argyfer gradd un</li> <li><a href="/wiki/Hafaliad_cwadratig" title="Hafaliad cwadratig">hafaliad cwadratig</a> ar gyfer gradd dau</li> <li>hafaliad ciwbig ar gyfer gradd tri</li> <li>hafaliad cwartig ar gyfer gradd pedwar</li> <li>hafaliad quintig ar gyfer gradd pump</li> <li>hafaliad sextig ar gyfer gradd chwech</li> <li>hafaliad septig ar gyfer gradd saith</li> <li>hafaliad octig ar gyfer gradd wyth</li></ul></li> <li>hafaliad Diophantine lle mae'n ofynnol i'r anhysbys fod yn <a href="/wiki/Cyfanrif" title="Cyfanrif">gyfanrifau</a></li> <li><a href="/w/index.php?title=Hafaliad_trosgynnol&action=edit&redlink=1" class="new" title="Hafaliad trosgynnol (dim tudalen ar gael)">hafaliad trosgynnol</a>, sy'n cynnwys swyddogaeth drosgynnol ei anhysbys</li> <li>hafaliad parametrig, lle mae'r datrysiadau ar gyfer y newidynnau yn cael eu mynegi fel swyddogaethau rhai newidynnau eraill, o'r enw <a href="/w/index.php?title=Paramedr&action=edit&redlink=1" class="new" title="Paramedr (dim tudalen ar gael)">paramedrau</a> sy'n ymddangos yn yr hafaliadau</li> <li>hafaliad swyddogaethol, lle mae'r anhysbys yn <a href="/wiki/Ffwythiant" title="Ffwythiant">ffwythiant</a> yn hytrach na meintiau syml</li> <li>hafaliadau sy'n cynnwys deilliadau, integrynnau a gwahaniaethau meidraidd ee: <ul><li><a href="/wiki/Hafaliad_differol" title="Hafaliad differol">hafaliad</a> <a href="/wiki/Hafaliad_differol" title="Hafaliad differol">differol</a></li> <li>hafaliad cyfannol (neu 'Integral equation')</li> <li>hafaliad swyddogaethol</li> <li>hafaliad <a href="https://en.wikipedia.org/wiki/Integro-differential_equation" class="extiw" title="en:Integro-differential equation">integro-differential</a></li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Cyfeiriadau">Cyfeiriadau</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hafaliad&veaction=edit&section=18" title="Golygu'r adran: Cyfeiriadau" class="mw-editsection-visualeditor"><span>golygu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hafaliad&action=edit&section=18" title="Edit section's source code: Cyfeiriadau"><span>golygu cod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist columns references-column-width" style="-moz-column-width: 30em; -webkit-column-width: 30em; column-width: 30em; list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><cite id="CITEREFLachaud" class="citation book">Lachaud, Gilles. "Équation, mathématique". <i>Encyclopædia Universalis</i> (yn Ffrangeg).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C3%89quation%2C+math%C3%A9matique&rft.btitle=Encyclop%C3%A6dia+Universalis&rft.aulast=Lachaud&rft.aufirst=Gilles&rfr_id=info%3Asid%2Fcy.wikipedia.org%3AHafaliad" class="Z3988"></span><style data-mw-deduplicate="TemplateStyles:r8312344">.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}</style></span> </li> <li id="cite_note-:1-2"><span class="mw-cite-backlink"><a href="#cite_ref-:1_2-0">↑</a></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://www.mathopenref.com/equation.html">"Equation - Math Open Reference"</a>. <i>www.mathopenref.com</i><span class="reference-accessdate">. Cyrchwyd <span class="nowrap">2020-09-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.mathopenref.com&rft.atitle=Equation+-+Math+Open+Reference&rft_id=https%3A%2F%2Fwww.mathopenref.com%2Fequation.html&rfr_id=info%3Asid%2Fcy.wikipedia.org%3AHafaliad" class="Z3988"></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r8312344"></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Weiner, Joan (2004).<i>Frege Explained</i>. Open Court.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Dolenni_allanol">Dolenni allanol</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hafaliad&veaction=edit&section=19" title="Golygu'r adran: Dolenni allanol" class="mw-editsection-visualeditor"><span>golygu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hafaliad&action=edit&section=19" title="Edit section's source code: Dolenni allanol"><span>golygu cod</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20090816161008/http://math.exeter.edu/rparris/winplot.html">Winplot</a> : Cynllwynwr Pwrpas Cyffredinol sy'n gallu darlunio ac animeiddio hafaliadau mathemategol 2D a 3D.</li> <li><a rel="nofollow" class="external text" href="http://www.cs.cornell.edu/w8/~andru/relplot">Cynllwynwr hafaliad</a> : Mae tudalen we ar gyfer cynhyrchu a lawrlwytho plotiau pdf neu ôl-nodyn o'r datrysiad yn gosod hafaliadau ac anghydraddoldebau mewn dau newidyn ( <i>x</i> ac <i>y</i> ).</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Wedi dod o "<a dir="ltr" href="https://cy.wikipedia.org/w/index.php?title=Hafaliad&oldid=12639750">https://cy.wikipedia.org/w/index.php?title=Hafaliad&oldid=12639750</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Arbennig:Categories" title="Arbennig:Categories">Categorïau</a>: <ul><li><a href="/wiki/Categori:Mathemateg_bur" title="Categori:Mathemateg bur">Mathemateg bur</a></li><li><a href="/wiki/Categori:Mathemateg_elfennol" title="Categori:Mathemateg elfennol">Mathemateg elfennol</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Chategori cudd: <ul><li><a href="/wiki/Categori:Tudalennau_gyda_meysydd_deublyg_yn_y_Nodion" title="Categori:Tudalennau gyda meysydd deublyg yn y Nodion">Tudalennau gyda meysydd deublyg yn y Nodion</a></li><li><a href="/wiki/Categori:CS1_Ffrangeg-language_sources_(fr)" title="Categori:CS1 Ffrangeg-language sources (fr)">CS1 Ffrangeg-language sources (fr)</a></li><li><a href="/wiki/Categori:Erthyglau_sy%27n_defnyddio_Nodyn_Gwybodlen_pethau_Wicidata" title="Categori:Erthyglau sy'n defnyddio Nodyn Gwybodlen pethau Wicidata">Erthyglau sy'n defnyddio Nodyn Gwybodlen pethau Wicidata</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"> Golygwyd y dudalen hon ddiwethaf ar 2 Mehefin 2024, am 15:45.</li> <li id="footer-info-copyright">Mae'r testun ar gael o dan <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-ShareAlike License</a>; gall telerau ychwanegol fod yn berthnasol. 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