{"id":342,"date":"2023-07-06T16:17:35","date_gmt":"2023-07-06T16:17:35","guid":{"rendered":"https:\/\/mathority.org\/nl\/jordan-gauss-methode-met-voorbeelden-en-opgeloste-oefeningen\/"},"modified":"2023-07-06T16:17:35","modified_gmt":"2023-07-06T16:17:35","slug":"jordan-gauss-methode-met-voorbeelden-en-opgeloste-oefeningen","status":"publish","type":"post","link":"https:\/\/mathority.org\/nl\/jordan-gauss-methode-met-voorbeelden-en-opgeloste-oefeningen\/","title":{"rendered":"Gaussische methode \u2013 jordani\u00eb"},"content":{"rendered":"<p>Op deze pagina leert u wat de Gauss-Jordan-methode is en hoe u een stelsel vergelijkingen kunt oplossen met behulp van de Gauss-methode. Daarnaast vind je ook voorbeelden en opgeloste oefeningen van systemen met de Gauss-methode zodat je deze perfect kunt oefenen en begrijpen.<\/p>\n<h2 class=\"wp-block-heading\"> Wat is de methode van Gauss?<\/h2>\n<p> De <strong>Gauss-Jordan-methode<\/strong> is een procedure die wordt gebruikt om stelsels vergelijkingen met 3 onbekenden op te lossen, dat wil zeggen als volgt:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-088146ef83bbd007e82aca8189434c25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{r} 3x-4y+5z=10 \\\\[2ex] x+5y-2z=4 \\\\[2ex] -x+4y+2z=-1 \\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"170\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Het doel van de methode van Gauss is om het initi\u00eble systeem van vergelijkingen om te zetten in een <strong>getrapt systeem<\/strong> , dat wil zeggen een systeem waarin elke vergelijking \u00e9\u00e9n minder onbekend heeft dan de vorige:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-10926b0856ae512c737ae924bd9413a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{r} a_1x+b_1y+c_1z=d_1 \\\\[2ex] a_2x+b_2y+c_2z=d_2 \\\\[2ex] a_3x+b_3y+c_3z=d_3 \\end{array} \\right\\} \\ \\bm{\\longrightarrow}   \\left. \\begin{array}{r} A_1x+B_1y+C_1z=D_1 \\\\[2ex] B_2y+C_2z=D_2 \\\\[2ex] C_3z=D_3 \\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"436\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Om dit te doen, moet u echter eerst weten hoe <strong>u een stelsel vergelijkingen in matrixvorm kunt uitdrukken<\/strong> en welke <strong>transformaties op deze matrix zijn toegestaan<\/strong> . We zullen deze twee dingen dus eerder uitleggen, en daarna zullen we zien hoe we de <strong>Gauss-methode<\/strong> kunnen gebruiken.<\/p>\n<h2 class=\"wp-block-heading\"> Systeem-uitgebreide matrix<\/h2>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Voordat je ziet hoe het systeem wordt opgelost, moet je weten dat <strong>een systeem van vergelijkingen kan worden uitgedrukt in de vorm van een matrix:<\/strong> de co\u00ebffici\u00ebnten van de<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-891f4f92ba63784e78eefc68d49377b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> worden in de eerste kolom geplaatst, de co\u00ebffici\u00ebnten van de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f731d739f7ff6bfef57b5a830dbe13aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> in de tweede kolom de co\u00ebffici\u00ebnten van de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> in de derde kolom en getallen zonder onbekenden in de vierde kolom.<\/p>\n<p> Bijvoorbeeld: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/resoudre-un-systeme-d-equations-par-la-methode-de-gauss.webp\" alt=\"Gaussische methode\" class=\"wp-image-4379\" width=\"422\" height=\"415\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"> Toegestane rijtransformaties<\/h2>\n<p> Om het stelsel vergelijkingen om te zetten in een geschaald systeem, kan een van de volgende bewerkingen worden uitgevoerd op de matrix die bij het systeem hoort:<\/p>\n<ul>\n<li> <strong><span style=\"color:#1976d2\">Wijzig de volgorde<\/span><\/strong> van rijen in de matrix.<\/li>\n<\/ul>\n<p> We kunnen bijvoorbeeld de volgorde van regels 2 en 3 van een matrix wijzigen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ee0e251559ef9dfd02c9b0105f934af8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} 3 &amp; 5 &amp; -2 &amp; 1 \\\\[2ex] -2 &amp; 4 &amp; -1 &amp; 2 \\\\[2ex] 6 &amp; 1 &amp; -3 &amp; 10 \\end{array} \\right)  \\begin{array}{c} \\\\[2ex] \\xrightarrow{ f_2 \\rightarrow f_3}} \\\\[2ex] \\xrightarrow{ f_3 \\rightarrow f_2}} \\end{array} \\left( \\begin{array}{ccc|c} 3 &amp; 5 &amp; -2 &amp; 1 \\\\[2ex] 6 &amp; 1 &amp; -3 &amp; 10 \\\\[2ex] -2 &amp; 4 &amp; -1 &amp; 2 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"98\" width=\"399\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong><span style=\"color:#1976d2\">Vermenigvuldig of deel<\/span><\/strong> alle termen in een rij met een ander getal dan 0.<\/li>\n<\/ul>\n<p> We kunnen bijvoorbeeld regel 1 vermenigvuldigen met 4 en regel 3 delen met 2:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0e1f081c9056075ede064b2e5c9e4193_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} 1 &amp; -2 &amp; 3 &amp; 1 \\\\[2ex] 3 &amp; -1 &amp; 5 &amp; -3 \\\\[2ex] 2 &amp; -4 &amp; -2 &amp; 6 \\end{array} \\right) \\begin{array}{c}  \\xrightarrow{4  f_1} \\\\[2ex]  \\\\[2ex] \\xrightarrow{ f_3 \/ 2} \\end{array} \\left( \\begin{array}{ccc|c} 4 &amp; -8 &amp; 12 &amp; 4 \\\\[2ex] 3 &amp; -1 &amp; 5 &amp; -3 \\\\[2ex] 1 &amp; -2 &amp; -1 &amp; 3 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"103\" width=\"396\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong><span style=\"color:#1976d2\">Vervang een rij<\/span><\/strong> door de som van dezelfde rij plus een andere rij vermenigvuldigd met een getal.<\/li>\n<\/ul>\n<p> In de volgende matrix voegen we bijvoorbeeld rij 2 toe aan rij 3, vermenigvuldigd met 1:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-04417e2094ac05c7a374334c55197f36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} -1 &amp; -3 &amp; 4 &amp; 1 \\\\[2ex] 2 &amp; 4 &amp; 1 &amp; -5 \\\\[2ex] 1 &amp; -2 &amp; 3 &amp; -1 \\end{array} \\right) \\begin{array}{c}   \\\\[2ex]  \\xrightarrow{f_2 + 1 \\cdot f_3}  \\\\[2ex] &amp; \\end{array} \\left( \\begin{array}{ccc|c} -1 &amp; -3 &amp; 4 &amp; 1 \\\\[2ex] 3 &amp; 2 &amp; 4 &amp; -6 \\\\[2ex] 1 &amp; -2 &amp; 3 &amp; -1 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"98\" width=\"417\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Hoe los je een stelsel vergelijkingen op met behulp van de Gauss-methode?<\/h2>\n<p> We zullen nu aan de hand van een voorbeeld de procedure bekijken <strong>voor het oplossen van een stelsel vergelijkingen met de Gauss-methode:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-61e6e829301e6730c9e27f9c0a30de2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{r} -x+2y+2z=-24 \\\\[2ex] x+y+z=48 \\\\[2ex] 2x-6y+4z=12 \\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"179\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Het eerste wat u moet doen is de <strong>uitgebreide matrix van het systeem<\/strong> : <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-de-systeme-dequations-par-la-methode-de-gauss.webp\" alt=\"Voorbeeld van een stelsel vergelijkingen opgelost met de Gauss-methode\" class=\"wp-image-913\" width=\"541\" height=\"175\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Zoals we later zullen zien, <strong>is het beter dat het eerste cijfer van de eerste regel een 1 is.<\/strong> We zullen daarom de volgorde van regels 1 en 2 veranderen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b45e0f757ca2880442314f6a4800697b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} -1 &amp; 2 &amp; 2 &amp;-24 \\\\[2ex] 1 &amp; 1 &amp; 1 &amp; 48 \\\\[2ex] 2 &amp; -6 &amp; 4 &amp; 12 \\end{array} \\right)  \\begin{array}{c} \\xrightarrow{ f_1 \\rightarrow f_2} \\\\[2ex] \\xrightarrow{ f_2 \\rightarrow f_1} \\\\[2ex] &amp; \\end{array} \\left( \\begin{array}{ccc|c}   \\color{blue}\\boxed{\\color{black}1} &amp; 1 &amp; 1 &amp; 48 \\\\[2ex] -1 &amp; 2 &amp; 2 &amp;-24 \\\\[2ex] 2 &amp; -6 &amp; 4 &amp; 12  \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"102\" width=\"496\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Het doel van de methode van Gauss is om <strong>de getallen onder de hoofddiagonaal 0<\/strong> te maken. Dat wil zeggen, we moeten de rode cijfers naar 0 converteren:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-28164ac6b48d32c09b4725548c0633f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} 1  &amp; 1 &amp; 1 &amp; 48 \\\\[2ex] \\color{red}\\bm{-1} &amp; 2 &amp; 2 &amp;-24 \\\\[2ex] \\color{red}\\bm{2} &amp; \\color{red}\\bm{-6} &amp; 4 &amp; 12  \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"224\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Om deze getallen te elimineren, moeten we de juiste transformaties van de rijen uitvoeren.<\/p>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<p> De -1, het eerste element van de tweede rij, is bijvoorbeeld het negatief van 1, het eerste element van de eerste rij. Als <strong>we daarom de eerste regel aan de tweede regel toevoegen,<\/strong> wordt de -1 ge\u00eblimineerd:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8b22cdc60e02d30a4ed31073c9ad47c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{lccc|c}  &amp; -1  &amp; 2 &amp; 2 &amp; -24  \\\\ + &amp; \\phantom{-}1  &amp; 1 &amp; 1 &amp; \\phantom{-}48   \\\\ \\hline &amp; \\phantom{-}0 &amp; 3 &amp; 3 &amp; \\phantom{-}24  \\end{array} \\begin{array}{l} \\color{blue}\\bm{\\leftarrow f_2} \\\\ \\color{blue}\\bm{\\leftarrow f_1} \\\\ \\phantom{hline} \\\\ \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"68\" width=\"245\" style=\"vertical-align: -29px;\"><\/p>\n<\/p>\n<p> Dus als we deze som berekenen, krijgen we de volgende matrix:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1b106306b92bfc3e99d602c22d5198bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} 1  &amp; 1 &amp; 1 &amp; 48 \\\\[2ex] -1 &amp; 2 &amp; 2 &amp; -24 \\\\[2ex] 2 &amp; -6 &amp; 4 &amp; 12 \\end{array} \\right) \\begin{array}{c}   \\\\[2ex]  \\xrightarrow{f_2 + f_1}  \\\\[2ex] &amp; \\end{array} \\left( \\begin{array}{ccc|c} 1  &amp; 1 &amp; 1 &amp; 48 \\\\[2ex]  \\color{blue}\\boxed{\\color{black}0} &amp; 3 &amp; 3 &amp; 24 \\\\[2ex] 2 &amp; -6 &amp; 4 &amp; 12 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"100\" width=\"479\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Op deze manier zijn we erin geslaagd om -1 om te zetten in een 0.<\/p>\n<p> Nu gaan we de 2 transformeren. Als je ziet, is de 2, het eerste element in de derde rij, het dubbele van 1, het eerste element in de eerste rij. Als <strong>we daarom de eerste rij vermenigvuldigd met -2 toevoegen aan de derde rij,<\/strong> wordt de 2 ge\u00eblimineerd:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-50716b37fe5c1ff45dc2e4b05300e3da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{lccc|c}    &amp;  \\phantom{-}2 &amp; -6 &amp; \\phantom{-}4 &amp; \\phantom{-}12  \\\\ + &amp; -2  &amp; -2 &amp; -2 &amp; -96 \\\\ \\hline &amp;  \\phantom{-}0 &amp; -8 &amp; \\phantom{-}2 &amp; -84  \\end{array} \\begin{array}{l} \\color{blue} \\bm{\\leftarrow f_3} \\\\ \\color{blue} \\bm{\\leftarrow -2 f_1} \\\\ \\phantom{hline} \\\\ \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"68\" width=\"295\" style=\"vertical-align: -29px;\"><\/p>\n<\/p>\n<p> We komen dus uit op de volgende matrix:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36b2fdf8de855cf35049ecefcf7c1da5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} 1  &amp; 1 &amp; 1 &amp; 48 \\\\[2ex]  0 &amp; 3 &amp; 3 &amp; 24 \\\\[2ex] 2 &amp; -6 &amp; 4 &amp; 12 \\end{array} \\right) \\begin{array}{c}   \\\\[2ex]    \\\\[2ex] \\xrightarrow{f_3-2f_1} \\end{array} \\left( \\begin{array}{ccc|c} 1  &amp; 1 &amp; 1 &amp; 48 \\\\[2ex]  0 &amp; 3 &amp; 3 &amp; 24 \\\\[2ex] \\color{blue}\\boxed{\\color{black}0} &amp; -8 &amp; 2 &amp; -84 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"100\" width=\"472\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Op deze manier zijn we erin geslaagd om de 2 om te zetten in een 0.<\/p>\n<p> Het enige wat we nu moeten doen is de -8 omzetten naar 0. Om dit te doen, <strong>vermenigvuldigen we de derde regel met 3 en voegen we de tweede regel vermenigvuldigd met 8 toe:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-156b06e383e387edd24cf4be09d98fe9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{lccc|r} &amp; 0  &amp; -24 &amp; \\phantom{2}6 &amp; -252  \\\\ + &amp; 0  &amp; \\phantom{-}24 &amp; 24 &amp; \\phantom{-}192  \\\\ \\hline  &amp; 0 &amp; \\phantom{-2}0 &amp; 30 &amp; -60  \\end{array} \\begin{array}{l}\\color{blue}\\bm{ \\leftarrow 3f_3} \\\\\\color{blue}\\bm{ \\leftarrow 8f_2} \\\\ \\phantom{hline} \\\\ \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"68\" width=\"281\" style=\"vertical-align: -29px;\"><\/p>\n<\/p>\n<p> We verkrijgen daarom de volgende matrix:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6e2324629222c746a9021ce05ba7d54d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} 1  &amp; 1 &amp; 1 &amp; 48 \\\\[2ex]  0 &amp; 3 &amp; 3 &amp; 24 \\\\[2ex] 0 &amp; -8 &amp; 2 &amp; -84 \\end{array} \\right) \\begin{array}{c}   \\\\[2ex]  \\\\[2ex] \\xrightarrow{3f_3 + 8f_2} \\end{array} \\left( \\begin{array}{ccc|c} 1  &amp; 1 &amp; 1 &amp; 48 \\\\[2ex]  0 &amp; 3 &amp; 3 &amp; 24 \\\\[2ex] 0 &amp; \\color{blue}\\boxed{\\color{black}0} &amp; 30 &amp; -60 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"100\" width=\"488\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> En met deze transformaties hebben we ervoor gezorgd dat <strong>alle getallen onder de hoofddiagonaal 0 zijn.<\/strong> Dus nu kunnen we het stelsel vergelijkingen oplossen.<\/p>\n<p> We moeten nu <strong>de matrix omzetten in een systeem van vergelijkingen met onbekenden<\/strong> . Om dit te doen, onthoud dat de eerste kolom overeenkomt met de<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> , de tweede kolom van<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> , de derde kolom van<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4586e340cb83d5b642972e97a288fec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> en de laatste kolom zijn de getallen zonder onbekenden:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6f90de9d9f5a06959a2d4aebf05f4758_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} 1  &amp; 1 &amp; 1 &amp; 48 \\\\[2ex]  0 &amp; 3 &amp; 3 &amp; 24 \\\\[2ex] 0 &amp; 0 &amp; 30 &amp; -60 \\end{array} \\right) \\  \\longrightarrow \\ \\left. \\begin{array}{r} 1x+1y+1z=48 \\\\[2ex] 3y+3z=24 \\\\[2ex] 30z=-60 \\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"379\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> En ten slotte moeten we, om het systeem op te lossen, <strong>de onbekenden van de vergelijkingen van onder naar boven oplossen.<\/strong> Omdat de laatste vergelijking slechts \u00e9\u00e9n onbekende heeft, kunnen we deze oplossen en de waarde ervan vinden: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f609bf44e2a363aca5008fec95c678dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"30z=-60\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"82\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-18c467d7caa0bb171c5b1590f9caa694_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z = \\cfrac{-60}{30}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"75\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9881bf11e418247b881ffbb7de1f0565_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{z=-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"55\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Nu we weten wat z is, kunnen we, als we de waarde ervan in de tweede vergelijking invullen, de waarde van vinden<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> : <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-47af5f5709e3e010efe9fb6f9ab0e8c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3y+3z=24 \\ \\xrightarrow{z \\ = \\ -2} \\ 3y+3(-2)=24\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"305\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-da13228341b35d3040ae42078cabd99a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3y-6=24\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"90\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8587739ff9abd17db5f6c4c51f4201dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3y=24+6\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"90\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8e6cb4d9575b53c3eafb342482f0dd28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3y=30\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d5e353e1b4b02441dfcb4258ee4b5480_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\cfrac{30}{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"53\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-819c20ffb8ea217c6f5fe196d891beec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y=10}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"51\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> En we doen hetzelfde met de eerste vergelijking: we vervangen de waarden van de andere onbekenden en we wissen<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> : <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-16197bd2e6cc012ef57ffb9e5dc77e45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1x+1y+1z=48 \\ \\xrightarrow{y \\ = \\ 10 \\ ; \\ z \\ = \\ -2} \\ 1x+1(10)+1(-2)=48\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"467\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a333e16025e12b5a3220e3673b32cfbc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x+10-2=48\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"122\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-11b343d96345b46d6686d9531c5ed39a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=48-10+2\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"121\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c64ae2e78dc27d215fff9629c45a07d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{x=40}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> De oplossing van het stelsel vergelijkingen is daarom: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-edf75d5dd0f035a459aaa69d22286a89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{x=40} \\quad \\bm{y=10} \\quad \\bm{z=-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"192\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-118\"><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"> Problemen van stelsels vergelijkingen opgelost volgens de Gauss-Jordan-methode<\/h2>\n<h3 class=\"wp-block-heading\"> Oefening 1<\/h3>\n<p> Los het volgende stelsel vergelijkingen op met behulp van de Gauss-methode: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-854043b0e7e3e2166593dcf5c645bfa0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{r} x+y-z=2 \\\\[2ex] x-2y+3z=0 \\\\[2ex] 2x-y+3z=3 \\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"143\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Het eerste dat we moeten doen is de uitgebreide matrix van het systeem:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1b6369a58b91f31bf4c8bc212ccf68c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{r} x+y-z=2 \\\\[2ex] x-2y+3z=0 \\\\[2ex] 2x-y+3z=3 \\end{array} \\right\\}  \\longrightarrow \\left( \\begin{array}{ccc|c} 1 &amp; 1 &amp; -1 &amp; 2 \\\\[2ex]  1 &amp; -2 &amp; 3 &amp; 0 \\\\[2ex] 2 &amp; -1 &amp; 3 &amp; 3 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"339\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu moeten we alle getallen onder de hoofdarray 0 maken.<\/p>\n<p class=\"has-text-align-left\"> We voeren daarom rijbewerkingen uit om de laatste twee termen van de eerste kolom te annuleren:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cd42dcf61aebc4c67de13e09dff72f4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c}1 &amp; 1 &amp; -1 &amp; 2 \\\\[2ex]  1 &amp; -2 &amp; 3 &amp; 0 \\\\[2ex] 2 &amp; -1 &amp; 3 &amp; 3 \\end{array} \\right) \\begin{array}{c} \\\\[2ex] \\xrightarrow{f_2 -f_1} \\\\[2ex] \\xrightarrow{f_3-2f_1} &amp; \\end{array} \\left( \\begin{array}{ccc|c} 1 &amp; 1 &amp; -1 &amp; 2 \\\\[2ex]  0 &amp; -3 &amp; 4 &amp; -2  \\\\[2ex] 0 &amp; -3 &amp; 5 &amp; -1 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"98\" width=\"399\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu verwijderen we het laatste element uit de tweede kolom:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-13945337848a6f1badf6efe249951124_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c}1 &amp; 1 &amp; -1 &amp; 2 \\\\[2ex]  0 &amp; -3 &amp; 4 &amp; -2 \\\\[2ex] 0 &amp; -3 &amp; 5 &amp; -1 \\end{array} \\right) \\begin{array}{c} \\\\[2ex]  \\\\[2ex] \\xrightarrow{f_3-f_2} &amp; \\end{array} \\left( \\begin{array}{ccc|c} 1 &amp; 1 &amp; -1 &amp; 2 \\\\[2ex] 0 &amp; -3 &amp; 4 &amp; -2 \\\\[2ex] 0 &amp; 0 &amp; 1 &amp; 1 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"98\" width=\"406\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Zodra alle getallen onder de hoofddiagonaal 0 zijn, kunnen we nu het stelsel vergelijkingen oplossen. Om dit te doen, drukken we de matrix opnieuw uit in de vorm van een stelsel van vergelijkingen met onbekenden:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f068c276aae018a668cc005bcad3e641_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} 1 &amp; 1 &amp; -1 &amp; 2 \\\\[2ex] 0 &amp; -3 &amp; 4 &amp; -2 \\\\[2ex] 0 &amp; 0 &amp; 1 &amp; 1 \\end{array} \\right) \\ \\longrightarrow \\ \\left. \\begin{array}{r} x+y-z=2 \\\\[2ex] -3y+4z=-2 \\\\[2ex] 1z=1 \\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"367\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En we lossen de onbekenden van de vergelijkingen van onder naar boven op. We lossen eerst de laatste vergelijking op: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-96b4633173cfb05c06e2c5bdc995d68c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1z= 1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"49\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-be8e058ccee5d0aa33ffcce09cd28f98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z=\\bm{1}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu vervangen we de waarde van z in de tweede vergelijking om de waarde van y te vinden: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-be96e507b80d44350c96abb013cdcaf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-3y+4z=-2 \\ \\xrightarrow{z \\ = \\ 1} \\ -3y+4(1)=-2\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"316\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8e9d92c48f937b8bffa678a761acee72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-3y+4=-2\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"107\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-23ff1832dcecc68a83d1e6afdf07a08a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-3y=-2-4\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"108\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-27cf5638a108b52a34c74053b099361e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-3y=-6\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1c9efb6ee00d7b62fd3f436c1037feb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\cfrac{-6}{-3} = \\bm{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"97\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En we doen hetzelfde met de eerste vergelijking: we vervangen de waarden van de andere onbekenden en we lossen op voor x: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-63f719497b296c95e6e9cf43651598c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x+y-z=2 \\ \\xrightarrow{y \\ = \\ 2 \\ ; \\ z \\ = \\ 1} \\  x+(2)-(1)=2\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"356\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-56845b4eb32748e1842b62d558249eb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x+1=2\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"72\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-02e03b55e65a6262280f3d3e592443ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=2-1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"72\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-df0486e1bf5773e392faebda4843f515_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{x=1}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> De oplossing van het stelsel vergelijkingen is daarom: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3acc67c77d6f5f92d9a684f87a5def90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{x=1} \\qquad \\bm{y=2} \\qquad \\bm{z=1}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"196\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Oefening 2<\/h3>\n<p> Vind de oplossing voor het volgende stelsel vergelijkingen met behulp van de Gauss-methode: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7d0595899b8137f769c74fce1b21286b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{r} 2x+y+2z=-3 \\\\[2ex] x+3y+2z=5 \\\\[2ex] 4x+2y-z=-1 \\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"157\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Het eerste dat we moeten doen is de uitgebreide matrix van het systeem:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5e2a16b6d1451520bd8898675c022dc2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{r} 2x+y+2z=-3 \\\\[2ex] x+3y+2z=5 \\\\[2ex] 4x+2y-z=-1 \\end{array} \\right\\}  \\longrightarrow \\left( \\begin{array}{ccc|c} 2 &amp; 1 &amp; 2 &amp; -3 \\\\[2ex] 1 &amp; 3 &amp; 2 &amp; 5 \\\\[2ex] 4 &amp; 2 &amp; -1 &amp; -1 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"352\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Om de Gauss-methode toe te passen is het eenvoudiger als het eerste getal op de eerste regel een 1 is. We zullen daarom de volgorde van regels 1 en 2 veranderen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef7e2e42d0eecb0395afb7c8311b2ade_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c}2 &amp; 1 &amp; 2 &amp; -3 \\\\[2ex] 1 &amp; 3 &amp; 2 &amp; 5 \\\\[2ex] 4 &amp; 2 &amp; -1 &amp; -1 \\end{array} \\right) \\begin{array}{c} \\xrightarrow{f_1\\rightarrow f_2} \\\\[2ex] \\xrightarrow{f_2\\rightarrow f_1} \\\\[2ex] &amp; \\end{array} \\left( \\begin{array}{ccc|c}1 &amp; 3 &amp; 2 &amp; 5 \\\\[2ex] 2 &amp; 1 &amp; 2 &amp; -3 \\\\[2ex]  4 &amp; 2 &amp; -1 &amp; -1\\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"101\" width=\"381\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu moeten we alle getallen onder de hoofdarray 0 maken.<\/p>\n<p class=\"has-text-align-left\"> We voeren dus rijbewerkingen uit om de laatste twee elementen van de eerste kolom te vervangen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-40baaee3bbde9ed1577e00bc1c3b338f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} 1 &amp; 3 &amp; 2 &amp; 5 \\\\[2ex] 2 &amp; 1 &amp; 2 &amp; -3 \\\\[2ex] 4 &amp; 2 &amp; -1 &amp; -1 \\end{array} \\right) \\begin{array}{c} \\\\[2ex] \\xrightarrow{f_2 -2f_1} \\\\[2ex] \\xrightarrow{f_3-4f_1} &amp; \\end{array} \\left( \\begin{array}{ccc|c} 1 &amp; 3 &amp; 2 &amp; 5 \\\\[2ex] 0 &amp; -5 &amp; -2 &amp; -13 \\\\[2ex] 0 &amp; -10 &amp; -9 &amp; -21 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"98\" width=\"417\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu converteren we het laatste element van de tweede kolom naar nul:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f328906485bfe6ee77833c04869e1240_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c}1 &amp; 3 &amp; 2 &amp; 5 \\\\[2ex] 0 &amp; -5 &amp; -2 &amp; -13 \\\\[2ex] 0 &amp; -10 &amp; -9 &amp; -21\\end{array} \\right) \\begin{array}{c} \\\\[2ex]  \\\\[2ex] \\xrightarrow{f_3-2f_2} &amp; \\end{array} \\left( \\begin{array}{ccc|c} 1 &amp; 3 &amp; 2 &amp; 5 \\\\[2ex] 0 &amp; -5 &amp; -2 &amp; -13 \\\\[2ex]  0 &amp; 0 &amp; -5 &amp; 5 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"98\" width=\"439\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Zodra alle getallen onder de hoofddiagonaal 0 zijn, kunnen we het stelsel vergelijkingen oplossen. Om dit te doen, drukken we de matrix opnieuw uit in de vorm van een stelsel van vergelijkingen met onbekenden:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7e129715c720218a5cb25ef07442442_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} 1 &amp; 3 &amp; 2 &amp; 5 \\\\[2ex] 0 &amp; -5 &amp; -2 &amp; -13 \\\\[2ex]  0 &amp; 0 &amp; -5 &amp; 5 \\end{array} \\right) \\ \\longrightarrow \\ \\left. \\begin{array}{r} x+3y+2z=5 \\\\[2ex] -5y-2z=-13 \\\\[2ex] -5z=5 \\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"384\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En we lossen de onbekenden van de vergelijkingen van onder naar boven op. We lossen eerst de laatste vergelijking op: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a52023bf513299dc6e11f3f0b83478cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-5z= 5\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"62\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8c401bac44e84469a254eca182dbaaf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z=\\cfrac{5}{-5}=\\bm{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"103\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu vervangen we de waarde van z in de tweede vergelijking om de waarde van y te vinden: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-802f76b9a10de545be30d16421b0a476_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-5y-2z=-13 \\ \\xrightarrow{z \\ = \\ -1} \\ -5y-2(-1)=-13\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"360\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dba702d28248bdc1928df387178232a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-5y+2=-13\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"117\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-00b2616ea47a24b0c533514cbad0074b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-5y=-13-2\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"116\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7f23dbeb21681cd1e0a85a6e980d578d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-5y=-15\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"86\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aa2c8d4e1526976ef2450ec835412a4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\cfrac{-15}{-5} = \\bm{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"107\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En we doen hetzelfde met de eerste vergelijking: we vervangen de waarden van de andere onbekenden en we lossen op voor x: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-664205a7f0b30720a191edc7ac6b5d4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x+3y+2z=5 \\ \\xrightarrow{y \\ = \\ 3 \\ ; \\ z \\ = \\ -1} \\  x+3(3)+2(-1)=5\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"416\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e08c94f818594a2e81185baa1b81c59e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x+9-2=5\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"103\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9e21aa5a70c92b8a72ef4c9a374d5acf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=5-9+2\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"103\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6505a1b32f86c9deb3ab0716f13c3949_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{x=-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> De oplossing van het stelsel vergelijkingen is daarom: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1be1c5f4156b37412c9e19a63190d45e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{x=-2} \\qquad \\bm{y=3} \\qquad \\bm{z=-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"224\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-119\"><\/div>\n<\/div>\n<h3 class=\"wp-block-heading\"> Oefening 3<\/h3>\n<p> Bereken de oplossing van het volgende stelsel vergelijkingen volgens de Gauss-methode: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4301eae3179543fbdee7568e8f88aa4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{r} 2x+3y+z=-1 \\\\[2ex] 6x+4y+4z=0 \\\\[2ex] -4x+2y-z=5 \\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"157\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Het eerste dat we moeten doen is de uitgebreide matrix van het systeem:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c0d96160d6670e817dd39f61816e1e6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{r} 2x+3y+z=-1 \\\\[2ex] 6x+4y+4z=0 \\\\[2ex] -4x+2y-z=5\\end{array} \\right\\}  \\longrightarrow \\left( \\begin{array}{ccc|c} 2 &amp; 3 &amp; 1 &amp; -1 \\\\[2ex] 6 &amp; 4 &amp; 4 &amp; 0 \\\\[2ex] -4 &amp; 2 &amp; -1 &amp; 5 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"366\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu moeten we alle getallen onder de bovenliggende array 0 maken.<\/p>\n<p class=\"has-text-align-left\"> We voeren dus rijbewerkingen uit om de laatste twee elementen van de eerste kolom te vervangen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-87853177b6be449178c24e414dc0865a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} 2 &amp; 3 &amp; 1 &amp; -1 \\\\[2ex] 6 &amp; 4 &amp; 4 &amp; 0 \\\\[2ex] -4 &amp; 2 &amp; -1 &amp; 5\\end{array} \\right) \\begin{array}{c} \\\\[2ex] \\xrightarrow{f_2 -3f_1} \\\\[2ex] \\xrightarrow{f_3+2f_1} &amp; \\end{array} \\left( \\begin{array}{ccc|c} 2 &amp; 3 &amp; 1 &amp; -1 \\\\[2ex] 0 &amp; -5 &amp; 1 &amp; 3 \\\\[2ex] 0 &amp; 8 &amp; 1 &amp; 3\\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"98\" width=\"399\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu converteren we het laatste element van de tweede kolom naar nul:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c4105ceb64b201c532109f8639bdefde_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c}2 &amp; 3 &amp; 1 &amp; -1 \\\\[2ex] 0 &amp; -5 &amp; 1 &amp; 3 \\\\[2ex] 0 &amp; 8 &amp; 1 &amp; 3\\end{array} \\right) \\begin{array}{c} \\\\[2ex]  \\\\[2ex] \\xrightarrow{5f_3+8f_2} &amp; \\end{array} \\left( \\begin{array}{ccc|c} 2 &amp; 3 &amp; 1 &amp; -1 \\\\[2ex] 0 &amp; -5 &amp; 1 &amp; 3 \\\\[2ex] 0 &amp; 0 &amp; 13 &amp; 39 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"98\" width=\"401\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Zodra alle getallen onder de hoofddiagonaal 0 zijn, kunnen we het stelsel vergelijkingen oplossen. Om dit te doen, drukken we de matrix opnieuw uit in de vorm van een stelsel van vergelijkingen met onbekenden:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-faae83295a3f7b3d8b6d76f78d56fac6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} 2 &amp; 3 &amp; 1 &amp; -1 \\\\[2ex] 0 &amp; -5 &amp; 1 &amp; 3 \\\\[2ex] 0 &amp; 0 &amp; 13 &amp; 39\\end{array} \\right) \\ \\longrightarrow \\ \\left. \\begin{array}{r} 2x+3y+1z=-1 \\\\[2ex] -5y+z=3 \\\\[2ex] 13z=39 \\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"389\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En we lossen de onbekenden van de vergelijkingen van onder naar boven op. We lossen eerst de laatste vergelijking op: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4ce4c9f218fa1bd1448e77039773f7d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"13z= 39\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"67\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2b17181241c9203cad7e9e776a3e4fbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z=\\cfrac{39}{13}=\\bm{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"85\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu vervangen we de waarde van z in de tweede vergelijking om de waarde van y te vinden: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-60d5784aa102e6db8696ba9bf79e1da5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-5y+z=3 \\ \\xrightarrow{z \\ = \\ 3} \\ -5y+(3)=3\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"272\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-48fbd56cf56ffcfb7d9c676eecf02550_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-5y=3-3\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"94\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8ee955981979845afcdca2dfbefe7fca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-5y=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"64\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-720b6716d584d822d06446bcc18382e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\cfrac{0}{-5} = \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"90\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En we doen hetzelfde met de eerste vergelijking: we vervangen de waarden van de andere onbekenden en we lossen op voor x: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ed92c6b5eec53094b5ff47ff6274f113_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x+3y+1z=-1 \\ \\xrightarrow{y \\ = \\ 0 \\ ; \\ z \\ = \\ 3} \\  2x+3(0)+1(3)=-1\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"437\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d7df18713df47399b523eb5025194be6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x+0+3=-1\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"126\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-994c82dc1b5d22d408db4477e0964fec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x=-1-3\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"96\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8bb8c1283df065c83c44b7fe484324a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x=-4\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"66\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d9dd0f99977b1208f94006aaa348d060_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\cfrac{-4}{2}=\\bm{-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"112\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> De oplossing van het stelsel vergelijkingen is daarom: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-93fd2b17430c25b9d44e5afc2e099e0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{x=-2} \\qquad \\bm{y=0} \\qquad \\bm{z=3}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"211\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Oefening 4<\/h3>\n<p> Los het volgende stelsel vergelijkingen met 3 onbekenden op met behulp van de Gauss-methode: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b005b2eda0d63c7130f2f5531c2ae4a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{r}  2x-6=4y+6z \\\\[2ex] -y-3z=1-3x \\\\[2ex] -4x-y=6-3z \\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"156\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Voordat we de methode van Gauss toepassen, moeten we het stelsel vergelijkingen zo rangschikken dat alle onbekenden zich aan de linkerkant van de vergelijking bevinden en de getallen aan de rechterkant:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1e0ca77b625e8f9e235ce8da4e4008df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{r}2x-6=4y+6z \\\\[2ex] -y-3z=1-3x \\\\[2ex] -4x-y=6-3z \\end{array} \\right\\} \\longrightarrow \\left.  \\begin{array}{r} 2x-4y-6z=6 \\\\[2ex] 3x-y-3z=1 \\\\[2ex] -4x-y+3z=6\\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"364\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Zodra het systeem is besteld, construeren we de ontwikkelde matrix van het systeem:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b88e3ff141b847028a55ba4b46b8e870_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{r} 2x-4y-6z=6 \\\\[2ex] 3x-y-3z=1 \\\\[2ex] -4x-y+3z=6 \\end{array} \\right\\}  \\longrightarrow \\left( \\begin{array}{ccc|c} 2 &amp; -4 &amp; -6 &amp; 6 \\\\[2ex] 3 &amp; -1 &amp; -3 &amp; 1 \\\\[2ex] -4 &amp; -1 &amp; 3 &amp; 6 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"365\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Omdat alle getallen in de eerste rij even zijn, delen we, voordat we met de rijen gaan werken, de eerste rij door 2. Omdat dit de berekeningen eenvoudiger maakt:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b05235526cd8e44c16749606bfe8976c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c}2 &amp; -4 &amp; -6 &amp; 6 \\\\[2ex] 3 &amp; -1 &amp; -3 &amp; 1 \\\\[2ex] -4 &amp; -1 &amp; 3 &amp; 6 \\end{array} \\right) \\begin{array}{c} \\xrightarrow{f_1\/2} \\\\[2ex] \\\\[2ex] &amp; \\end{array} \\left( \\begin{array}{ccc|c}1 &amp; -2 &amp; -3 &amp; 3 \\\\[2ex] 3 &amp; -1 &amp; -3 &amp; 1 \\\\[2ex] -4 &amp; -1 &amp; 3 &amp; 6\\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"99\" width=\"396\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu moeten we alle getallen onder de hoofdarray 0 maken.<\/p>\n<p class=\"has-text-align-left\"> We voeren dus rijbewerkingen uit om de laatste twee elementen van de eerste kolom te vervangen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3da82815d14fdfae0f61a8e1747fb9fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} 1 &amp; -2 &amp; -3 &amp; 3 \\\\[2ex] 3 &amp; -1 &amp; -3 &amp; 1 \\\\[2ex] -4 &amp; -1 &amp; 3 &amp; 6 \\end{array} \\right) \\begin{array}{c} \\\\[2ex] \\xrightarrow{f_2 -3f_1} \\\\[2ex] \\xrightarrow{f_3+4f_1} &amp; \\end{array} \\left( \\begin{array}{ccc|c} 1 &amp; -2 &amp; -3 &amp; 3 \\\\[2ex] 0 &amp; 5 &amp; 6 &amp; -8 \\\\[2ex] 0 &amp; -9 &amp; -9 &amp; 18\\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"98\" width=\"413\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Omdat alle getallen in de laatste regel een veelvoud van 9 zijn, delen we dit net als voorheen door 9 om de berekeningen eenvoudiger te maken:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-342000d19a7bd19e055a39695c79cb49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c}1 &amp; -2 &amp; -3 &amp; 3 \\\\[2ex] 0 &amp; 5 &amp; 6 &amp; -8 \\\\[2ex] 0 &amp; -9 &amp; -9 &amp; 18 \\end{array} \\right) \\begin{array}{c}  \\\\[2ex] \\\\[2ex]\\xrightarrow{f_3\/9} &amp; \\end{array} \\left( \\begin{array}{ccc|c}1 &amp; -2 &amp; -3 &amp; 3 \\\\[2ex] 0 &amp; 5 &amp; 6 &amp; -8 \\\\[2ex] 0 &amp; -1 &amp; -1 &amp; 2\\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"396\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu converteren we het laatste element van de tweede kolom naar nul:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2158e7f439f677617bb8a40695fb5711_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c}1 &amp; -2 &amp; -3 &amp; 3 \\\\[2ex] 0 &amp; 5 &amp; 6 &amp; -8 \\\\[2ex] 0 &amp; -1 &amp; -1 &amp; 2\\end{array} \\right) \\begin{array}{c} \\\\[2ex]  \\\\[2ex] \\xrightarrow{5f_3+f_2} &amp; \\end{array} \\left( \\begin{array}{ccc|c} 1 &amp; -2 &amp; -3 &amp; 3 \\\\[2ex] 0 &amp; 5 &amp; 6 &amp; -8 \\\\[2ex] 0 &amp; 0 &amp; 1 &amp; 2 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"98\" width=\"413\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Zodra alle getallen onder de hoofddiagonaal 0 zijn, kunnen we het stelsel vergelijkingen oplossen. Om dit te doen, drukken we de matrix opnieuw uit in de vorm van een stelsel van vergelijkingen met onbekenden:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ea162e98aa70f8d56ffba28438a9de2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{array}{ccc|c} 1 &amp; -2 &amp; -3 &amp; 3 \\\\[2ex] 0 &amp; 5 &amp; 6 &amp; -8 \\\\[2ex] 0 &amp; 0 &amp; 1 &amp; 2 \\end{array} \\right) \\ \\longrightarrow \\ \\left. \\begin{array}{r} x-2y-3z=3 \\\\[2ex] 5y+6z=-8 \\\\[2ex] 1z=2 \\end{array} \\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"371\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En we lossen de onbekenden van de vergelijkingen van onder naar boven op. We lossen eerst de laatste vergelijking op: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-28e1aa243598bedb00978eb5e0c1dcd3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1z= 2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"49\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4307991ada04e86ea4085fe426ea9f08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"z=\\bm{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu vervangen we de waarde van z in de tweede vergelijking om de waarde van y te vinden: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b5eb143e26fb1f1535d4c2596c889007_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"5y+6z=-8 \\ \\xrightarrow{z \\ = \\ 2} \\ 5y+6(2)=-8\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"291\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-48b6461c0e3832049edb903a568752df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"5y+12=-8\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"104\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4c9094946053033193fa68d290041f3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"5y=-8-12\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"103\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8492d7042878af6f3863f5e83213a2ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"5y=-20\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"74\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fe3119924b2e0ee7a98cc5489223e689_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=\\cfrac{-20}{5} = \\bm{-4}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"121\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En we doen hetzelfde met de eerste vergelijking: we vervangen de waarden van de andere onbekenden en we lossen op voor x: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d3e3bbe943253ea68f811850c1b882bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x-2y-3z=3 \\ \\xrightarrow{y \\ = \\ -4 \\ ; \\ z \\ = \\ 2} \\  x-2(-4)-3(2)=3\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"417\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b5d555ad593794ad6d247dbbc2cd98eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x+8-6=3\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"104\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-af64e62e2b2b22939ce0f08900f91404_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=3-8+6\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"104\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-df0486e1bf5773e392faebda4843f515_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{x=1}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> De oplossing van het stelsel vergelijkingen is daarom:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-586cec7145ab5b293cadaafd4f7bb738_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{x=-1} \\qquad \\bm{y=-4} \\qquad \\bm{z=2}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"224\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Op deze pagina leert u wat de Gauss-Jordan-methode is en hoe u een stelsel vergelijkingen kunt oplossen met behulp van de Gauss-methode. Daarnaast vind je ook voorbeelden en opgeloste oefeningen van systemen met de Gauss-methode zodat je deze perfect kunt oefenen en begrijpen. Wat is de methode van Gauss? De Gauss-Jordan-methode is een procedure die &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/nl\/jordan-gauss-methode-met-voorbeelden-en-opgeloste-oefeningen\/\"> <span class=\"screen-reader-text\">Gaussische methode \u2013 jordani\u00eb<\/span> Lees meer &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[41],"tags":[],"class_list":["post-342","post","type-post","status-publish","format-standard","hentry","category-wiskundige-verklaringen"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Gaussische methode \u2013 Jordani\u00eb -<\/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:\/\/mathority.org\/nl\/jordan-gauss-methode-met-voorbeelden-en-opgeloste-oefeningen\/\" \/>\n<meta property=\"og:locale\" content=\"nl_NL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Gaussische methode \u2013 Jordani\u00eb -\" \/>\n<meta property=\"og:description\" content=\"Op deze pagina leert u wat de Gauss-Jordan-methode is en hoe u een stelsel vergelijkingen kunt oplossen met behulp van de Gauss-methode. Daarnaast vind je ook voorbeelden en opgeloste oefeningen van systemen met de Gauss-methode zodat je deze perfect kunt oefenen en begrijpen. Wat is de methode van Gauss? 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