{"id":79,"date":"2021-09-10T11:20:42","date_gmt":"2021-09-10T09:20:42","guid":{"rendered":"https:\/\/victorbrunet.fr\/?p=79"},"modified":"2021-09-10T11:20:42","modified_gmt":"2021-09-10T09:20:42","slug":"methode-du-pivot-de-gauss","status":"publish","type":"post","link":"https:\/\/victorbrunet.fr\/index.php\/2021\/09\/10\/methode-du-pivot-de-gauss\/","title":{"rendered":"M\u00e9thode du pivot de Gauss"},"content":{"rendered":"<p class=\"dtexte\">La m\u00e9thode du pivot de Gauss est une m\u00e9thode pour transformer un syst\u00e8me en un autre syst\u00e8me \u00e9quivalent (ayant les m\u00eames solutions) qui est triangulaire et est donc facile \u00e0 r\u00e9soudre. Les op\u00e9rations autoris\u00e9es pour transformer ce syst\u00e8me sont :<\/p>\n<div class=\"liste listetexte\">\n<ul>\n<li>\u00e9change de deux lignes.<\/li>\n<li>multiplication d&rsquo;une ligne par un nombre non nul.<\/li>\n<li>addition d&rsquo;un multiple d&rsquo;une ligne \u00e0 une autre ligne.<\/li>\n<\/ul>\n<\/div>\n<p class=\"dtexte\">Prenons l&rsquo;exemple suivant :<\/p>\n<p><center><img decoding=\"async\" src=\"https:\/\/www.bibmath.net\/dico\/g\/images\/gausspivot1.png\" \/><\/center><\/p>\n<p class=\"ntexte\">On conserve la ligne L<sub>1<\/sub>, qui sert de pivot pour \u00e9liminer l&rsquo;inconnue\u00a0<span id=\"MathJax-Element-2-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"box-sizing: border-box; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 18.656px; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;x&lt;\/mi&gt;&lt;\/math&gt;\"><span id=\"MJXc-Node-3\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-4\" class=\"mjx-mrow\"><span id=\"MJXc-Node-5\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">x<\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">x<\/span><\/span>\u00a0des autres lignes; pour cela, on retire L<sub>1<\/sub>\u00a0\u00e0 L<sub>2<\/sub>, et 3 fois L<sub>1<\/sub>\u00a0\u00e0 L<sub>3<\/sub>. On obtient :<\/p>\n<p><center><img decoding=\"async\" src=\"https:\/\/www.bibmath.net\/dico\/g\/images\/gausspivot2.png\" \/><\/center><\/p>\n<p class=\"ntexte\">On conserve alors la ligne L<sub>2<\/sub>\u00a0qui sert de pivot pour \u00e9liminer\u00a0<span id=\"MathJax-Element-3-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"box-sizing: border-box; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 18.656px; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;y&lt;\/mi&gt;&lt;\/math&gt;\"><span id=\"MJXc-Node-6\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-7\" class=\"mjx-mrow\"><span id=\"MJXc-Node-8\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">y<\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">y<\/span><\/span>\u00a0de la troisi\u00e8me ligne; pour cela, on remplace la ligne L<sub>3<\/sub>\u00a0par L<sub>3<\/sub>+L<sub>2<\/sub>. On trouve :<\/p>\n<p><center><img decoding=\"async\" src=\"https:\/\/www.bibmath.net\/dico\/g\/images\/gausspivot3.png\" \/><\/center><\/p>\n<p class=\"ntexte\">Ce dernier syst\u00e8me, triangulaire, est facile \u00e0 r\u00e9soudre : la derni\u00e8re ligne donne\u00a0<span id=\"MathJax-Element-4-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"box-sizing: border-box; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 18.656px; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;z&lt;\/mi&gt;&lt;\/math&gt;\"><span id=\"MJXc-Node-9\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-10\" class=\"mjx-mrow\"><span id=\"MJXc-Node-11\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">z<\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">z<\/span><\/span>, en reportant, la deuxi\u00e8me ligne donne\u00a0<span id=\"MathJax-Element-5-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"box-sizing: border-box; display: inline-block; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 18.656px; letter-spacing: normal; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;y&lt;\/mi&gt;&lt;\/math&gt;\"><span id=\"MJXc-Node-12\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-13\" class=\"mjx-mrow\"><span id=\"MJXc-Node-14\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">y<\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">y<\/span><\/span>, etc&#8230;<\/p>\n<p><em>Source : https:\/\/www.bibmath.net\/dico\/index.php?action=affiche&amp;quoi=.\/g\/gausspivot.html<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>La m\u00e9thode du pivot de Gauss est une m\u00e9thode pour transformer un syst\u00e8me en un autre syst\u00e8me \u00e9quivalent (ayant les m\u00eames solutions) qui est triangulaire et est donc facile \u00e0 r\u00e9soudre. Les op\u00e9rations autoris\u00e9es pour transformer ce syst\u00e8me sont : \u00e9change de deux lignes. multiplication d&rsquo;une ligne par un nombre non nul. addition d&rsquo;un multiple [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":80,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-79","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-math"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v24.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>M\u00e9thode du pivot de Gauss - Victor BRUNET<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/victorbrunet.fr\/index.php\/2021\/09\/10\/methode-du-pivot-de-gauss\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"M\u00e9thode du pivot de Gauss - Victor BRUNET\" \/>\n<meta property=\"og:description\" content=\"La m\u00e9thode du pivot de Gauss est une m\u00e9thode pour transformer un syst\u00e8me en un autre syst\u00e8me \u00e9quivalent (ayant les m\u00eames solutions) qui est triangulaire et est donc facile \u00e0 r\u00e9soudre. 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