{"id":1477,"date":"2025-07-25T02:06:06","date_gmt":"2025-07-25T02:06:06","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1477"},"modified":"2026-03-11T09:25:56","modified_gmt":"2026-03-11T09:25:56","slug":"solving-systems-with-cramers-rule-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/solving-systems-with-cramers-rule-fresh-take\/","title":{"raw":"Solving Systems with Cramer\u2019s Rule: Fresh Take","rendered":"Solving Systems with Cramer\u2019s Rule: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Evaluate 2 \u00d7 2 and 3 \u00d7 3 determinants<\/li>\r\n \t<li>Use Cramer's Rule to solve a system of equations in two variables<\/li>\r\n \t<li>Use Cramer's Rule to solve a system of three equations in three variables<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Evaluate 2 \u00d7 2 and 3 \u00d7 3 Determinants<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Determinant:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">A real number calculated from a square matrix<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Notation: [latex]\\det(A)[\/latex] or [latex]|A|[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Used to determine if a matrix is invertible<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">2 \u00d7 2 Determinant Formula:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For [latex]A = \\left[\\begin{array}{cc}a &amp; b\\\\ c &amp; d\\end{array}\\right][\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\det(A) = ad - bc[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>3 x 3 Determinants: use calculator function det()<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For 2 \u00d7 2 Matrices:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Multiply diagonal entries (top-left to bottom-right)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Subtract product of other diagonal (top-right to bottom-left)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For 3 \u00d7 3 Matrices:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Augment matrix by repeating first two columns<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Add products of three diagonals going down-right<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Subtract products of three diagonals going up-right<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Find the determinant of [latex]A = \\left[\\begin{array}{cc}5 &amp; 2\\\\ -6 &amp; 3\\end{array}\\right][\/latex].[reveal-answer q=\"det-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"det-001\"]\r\n[latex]\\begin{align*}\r\n\\det(A) &amp;= \\left\\rvert\\begin{array}{cc}5 &amp; 2\\\\ -6 &amp; 3\\end{array}\\right\\rvert \\\\\r\n&amp;= 5(3) - (-6)(2) \\\\\r\n&amp;= 15 + 12 \\\\\r\n&amp;= 27\r\n\\end{align*}[\/latex]\r\n[\/hidden-answer]<\/section><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gagghcfc-OU9sWHk_dlw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/OU9sWHk_dlw?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gagghcfc-OU9sWHk_dlw\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660455&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-gagghcfc-OU9sWHk_dlw&vembed=0&video_id=OU9sWHk_dlw&video_target=tpm-plugin-gagghcfc-OU9sWHk_dlw'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Finding+the+determinant+of+a+2x2+matrix+%7C+Matrices+%7C+Precalculus+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFinding the determinant of a 2x2 matrix | Matrices | Precalculus | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]315405[\/ohm_question]<\/section><section class=\"textbox example\">Find the determinant of [latex]A = \\left[\\begin{array}{ccc}0 &amp; 2 &amp; 1\\\\ 3 &amp; -1 &amp; 1\\\\ 4 &amp; 0 &amp; 1\\end{array}\\right][\/latex].[reveal-answer q=\"det-002\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"det-002\"]\r\nAugment with the first two columns:[latex]\\left\\rvert\\begin{array}{ccc}0 &amp; 2 &amp; 1\\\\ 3 &amp; -1 &amp; 1\\\\ 4 &amp; 0 &amp; 1\\end{array}\\right\\rvert\\left.\\begin{array}{cc}0 &amp; 2\\\\ 3 &amp; -1\\\\ 4 &amp; 0\\end{array}\\right\\rvert[\/latex]Calculate:Starting from the top:\r\n<ul>\r\n \t<li>The first diagonal contains the entries: [latex]0,-1,1[\\latex]<\/li>\r\n \t<li>The second diagonal contains the entries: [latex]2,1,4[\\latex]<\/li>\r\n \t<li>The third\u00a0 diagonal contains the entries: [latex]1,3,0[\\latex]<\/li>\r\n<\/ul>\r\n[latex]0(-1)(1) + 2(1)(4) + 1(3)(0) = 8[latex]\r\n<ul>\r\n \t<li>The first diagonal contains the entries: [latex]4,-1,1[\\latex]<\/li>\r\n \t<li>The second diagonal contains the entries: [latex]0,1,0[\\latex]<\/li>\r\n \t<li>The third\u00a0 diagonal contains the entries: [latex]1,3,2[\\latex]<\/li>\r\n<\/ul>\r\n[latex]4(-1)(1) + 0(1)(0) + 1(3)(2) = 2[latex]\r\n\r\nSubtract the bottom diagonals from the top diagonals:\r\n\r\n[latex]8-2=6[\/latex]\r\n\r\n[latex]|A|=6[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gcbcecbh-R7BWKnO6xNY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/R7BWKnO6xNY?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gcbcecbh-R7BWKnO6xNY\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660456&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-gcbcecbh-R7BWKnO6xNY&vembed=0&video_id=R7BWKnO6xNY&video_target=tpm-plugin-gcbcecbh-R7BWKnO6xNY'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Determinant+of+a+3x3+matrix+using+Augmented+Matrices_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDeterminant of a 3x3 matrix using Augmented Matrices\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]315407[\/ohm_question]<\/section><section><section class=\"textbox watchIt\" aria-label=\"Watch It\">[videopicker divId=\"tnh-video-picker\" title=\"Choose a Calculator\" label=\"Choose Calculator\"]\r\n[videooption displayName=\"TI-84\" value=\"\/\/plugin.3playmedia.com\/show?mf=14660457&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=Q-Dv6V2G47c&amp;video_target=tpm-plugin-drvnscs7-Q-Dv6V2G47c\"][videooption displayName=\"Desmos Matrix Calculator\" value=\"\/\/plugin.3playmedia.com\/show?mf=14660458&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=pXtGyKbD5uY&amp;video_target=tpm-plugin-drvnscs7-pXtGyKbD5uY\"]\r\n[\/videopicker]\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Calculating+the+Determinant+of+a+Matrix+(TI+84+Plus+CE)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCalculating the Determinant of a Matrix (TI 84 Plus CE)\u201d here (opens in new window).<\/a>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/DESMOS+find+determinant_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDESMOS find determinant\u201d here (opens in new window).<\/a><\/section><\/section>\r\n<h2>Use Cramer's Rule to Solve Systems in Two Variables<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Cramer's Rule:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Method using determinants to solve systems<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Works when number of equations equals number of variables<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]D = 0[\/latex], system has no solution or infinite solutions<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For system [latex]\\begin{array}{l}{a}_{1}x+{b}_{1}y={c}_{1}\\\\ {a}_{2}x+{b}_{2}y={c}_{2}\\end{array}[\/latex]:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]D = \\left\\rvert\\begin{array}{cc}{a}_{1} &amp; {b}_{1}\\\\ {a}_{2} &amp; {b}_{2}\\end{array}\\right\\rvert[\/latex] (coefficient matrix)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]D_x = \\left\\rvert\\begin{array}{cc}{c}_{1} &amp; {b}_{1}\\\\ {c}_{2} &amp; {b}_{2}\\end{array}\\right\\rvert[\/latex] (replace x-column with constants)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]D_y = \\left\\rvert\\begin{array}{cc}{a}_{1} &amp; {c}_{1}\\\\ {a}_{2} &amp; {c}_{2}\\end{array}\\right\\rvert[\/latex] (replace y-column with constants)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]x = \\frac{D_x}{D}[\/latex], [latex]y = \\frac{D_y}{D}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Find [latex]D[\/latex] (determinant of coefficient matrix)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find [latex]D_x[\/latex] (replace x-column with constant column)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find [latex]D_y[\/latex] (replace y-column with constant column)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate [latex]x = \\frac{D_x}{D}[\/latex] and [latex]y = \\frac{D_y}{D}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Solve using Cramer's Rule: [latex]\\begin{array}{l}12x + 3y = 15\\\\ 2x - 3y = 13\\end{array}[\/latex][reveal-answer q=\"cramer-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"cramer-001\"]Find [latex]D[\/latex]:[latex]D = \\left\\rvert\\begin{array}{cc}12 &amp; 3\\\\ 2 &amp; -3\\end{array}\\right\\rvert = 12(-3) - 2(3) = -36 - 6 = -42[\/latex]Find [latex]D_x[\/latex]:\r\n[latex]D_x = \\left\\rvert\\begin{array}{cc}15 &amp; 3\\\\ 13 &amp; -3\\end{array}\\right\\rvert = 15(-3) - 13(3) = -45 - 39 = -84[\/latex]Find [latex]D_y[\/latex]:\r\n[latex]D_y = \\left\\rvert\\begin{array}{cc}12 &amp; 15\\\\ 2 &amp; 13\\end{array}\\right\\rvert = 12(13) - 2(15) = 156 - 30 = 126[\/latex]Solve:\r\n[latex]x = \\frac{D_x}{D} = \\frac{-84}{-42} = 2[\/latex][latex]y = \\frac{D_y}{D} = \\frac{126}{-42} = -3[\/latex]The solution is [latex](2, -3)[\/latex].\r\n[\/hidden-answer]<\/section><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dggdhgae-Yr9hTPvl8Ng\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Yr9hTPvl8Ng?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dggdhgae-Yr9hTPvl8Ng\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660459&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dggdhgae-Yr9hTPvl8Ng&vembed=0&video_id=Yr9hTPvl8Ng&video_target=tpm-plugin-dggdhgae-Yr9hTPvl8Ng'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Learn+How+to+Use+Cramer's+Rule+to+Solve+a+2+x+2+System+of+Equations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLearn How to Use Cramer's Rule to Solve a 2 x 2 System of Equations\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]315410[\/ohm_question]<\/section>\r\n<h2>Use Cramer's Rule to Solve Systems in Three Variables<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">For 3 \u00d7 3 Systems:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Same pattern as 2 \u00d7 2 systems<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Replace each variable's column with constants to find that variable<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]x = \\frac{D_x}{D}[\/latex], [latex]y = \\frac{D_y}{D}[\/latex], [latex]z = \\frac{D_z}{D}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Find [latex]D[\/latex] (determinant of coefficient matrix)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find [latex]D_x[\/latex], [latex]D_y[\/latex], and [latex]D_z[\/latex] by replacing appropriate columns<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate each variable using the formulas<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Solve using Cramer's Rule:\r\n[latex]\\begin{array}{l}x + y - z = 6\\\\ 3x - 2y + z = -5\\\\ x + 3y - 2z = 14\\end{array}[\/latex][reveal-answer q=\"cramer-002\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"cramer-002\"]Set up determinants:[latex]D = \\left\\rvert\\begin{array}{ccc}1 &amp; 1 &amp; -1\\\\ 3 &amp; -2 &amp; 1\\\\ 1 &amp; 3 &amp; -2\\end{array}\\right\\rvert = -3[\/latex][latex]D_x = \\left\\rvert\\begin{array}{ccc}6 &amp; 1 &amp; -1\\\\ -5 &amp; -2 &amp; 1\\\\ 14 &amp; 3 &amp; -2\\end{array}\\right\\rvert = -3[\/latex][latex]D_y = \\left\\rvert\\begin{array}{ccc}1 &amp; 6 &amp; -1\\\\ 3 &amp; -5 &amp; 1\\\\ 1 &amp; 14 &amp; -2\\end{array}\\right\\rvert = -9[\/latex][latex]D_z = \\left\\rvert\\begin{array}{ccc}1 &amp; 1 &amp; 6\\\\ 3 &amp; -2 &amp; -5\\\\ 1 &amp; 3 &amp; 14\\end{array}\\right\\rvert = 6[\/latex]Solve:\r\n[latex]x = \\frac{-3}{-3} = 1[\/latex], [latex]y = \\frac{-9}{-3} = 3[\/latex], [latex]z = \\frac{6}{-3} = -2[\/latex]The solution is [latex](1, 3, -2)[\/latex].\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cbhdeafd-ziMrl-31PfY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ziMrl-31PfY?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cbhdeafd-ziMrl-31PfY\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660460&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-cbhdeafd-ziMrl-31PfY&vembed=0&video_id=ziMrl-31PfY&video_target=tpm-plugin-cbhdeafd-ziMrl-31PfY'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+do+you+use+Cramer%E2%80%99s+Rule+to+solve+Systems+of+3+Linear+Equations%3F+The+Easiest+Method!_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow do you use Cramer\u2019s Rule to solve Systems of 3 Linear Equations? The Easiest Method!\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Evaluate 2 \u00d7 2 and 3 \u00d7 3 determinants<\/li>\n<li>Use Cramer&#8217;s Rule to solve a system of equations in two variables<\/li>\n<li>Use Cramer&#8217;s Rule to solve a system of three equations in three variables<\/li>\n<\/ul>\n<\/section>\n<h2>Evaluate 2 \u00d7 2 and 3 \u00d7 3 Determinants<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Determinant:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">A real number calculated from a square matrix<\/li>\n<li class=\"whitespace-normal break-words\">Notation: [latex]\\det(A)[\/latex] or [latex]|A|[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Used to determine if a matrix is invertible<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">2 \u00d7 2 Determinant Formula:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]A = \\left[\\begin{array}{cc}a & b\\\\ c & d\\end{array}\\right][\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\det(A) = ad - bc[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>3 x 3 Determinants: use calculator function det()<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For 2 \u00d7 2 Matrices:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Multiply diagonal entries (top-left to bottom-right)<\/li>\n<li class=\"whitespace-normal break-words\">Subtract product of other diagonal (top-right to bottom-left)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">For 3 \u00d7 3 Matrices:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Augment matrix by repeating first two columns<\/li>\n<li class=\"whitespace-normal break-words\">Add products of three diagonals going down-right<\/li>\n<li class=\"whitespace-normal break-words\">Subtract products of three diagonals going up-right<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Find the determinant of [latex]A = \\left[\\begin{array}{cc}5 & 2\\\\ -6 & 3\\end{array}\\right][\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qdet-001\">Show Solution<\/button><\/p>\n<div id=\"qdet-001\" class=\"hidden-answer\" style=\"display: none\">\n[latex]\\begin{align*}  \\det(A) &= \\left\\rvert\\begin{array}{cc}5 & 2\\\\ -6 & 3\\end{array}\\right\\rvert \\\\  &= 5(3) - (-6)(2) \\\\  &= 15 + 12 \\\\  &= 27  \\end{align*}[\/latex]\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gagghcfc-OU9sWHk_dlw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/OU9sWHk_dlw?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gagghcfc-OU9sWHk_dlw\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14660455&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-gagghcfc-OU9sWHk_dlw&#38;vembed=0&#38;video_id=OU9sWHk_dlw&#38;video_target=tpm-plugin-gagghcfc-OU9sWHk_dlw\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Finding+the+determinant+of+a+2x2+matrix+%7C+Matrices+%7C+Precalculus+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFinding the determinant of a 2&#215;2 matrix | Matrices | Precalculus | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm315405\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=315405&theme=lumen&iframe_resize_id=ohm315405&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">Find the determinant of [latex]A = \\left[\\begin{array}{ccc}0 & 2 & 1\\\\ 3 & -1 & 1\\\\ 4 & 0 & 1\\end{array}\\right][\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qdet-002\">Show Solution<\/button><\/p>\n<div id=\"qdet-002\" class=\"hidden-answer\" style=\"display: none\">\nAugment with the first two columns:[latex]\\left\\rvert\\begin{array}{ccc}0 & 2 & 1\\\\ 3 & -1 & 1\\\\ 4 & 0 & 1\\end{array}\\right\\rvert\\left.\\begin{array}{cc}0 & 2\\\\ 3 & -1\\\\ 4 & 0\\end{array}\\right\\rvert[\/latex]Calculate:Starting from the top:<\/p>\n<ul>\n<li>The first diagonal contains the entries: [latex]0,-1,1[\\latex]<\/li>\n<li>The second diagonal contains the entries: [latex]2,1,4[\\latex]<\/li>\n<li>The third\u00a0 diagonal contains the entries: [latex]1,3,0[\\latex]<\/li>\n<\/ul>\n<p>  [latex]0(-1)(1) + 2(1)(4) + 1(3)(0) = 8[latex]  <\/p>\n<ul>\n<li>The first diagonal contains the entries: [latex]4,-1,1[\\latex]<\/li>\n<li>The second diagonal contains the entries: [latex]0,1,0[\\latex]<\/li>\n<li>The third\u00a0 diagonal contains the entries: [latex]1,3,2[\\latex]<\/li>\n<\/ul>\n<p>  [latex]4(-1)(1) + 0(1)(0) + 1(3)(2) = 2[latex]    Subtract the bottom diagonals from the top diagonals:    [latex]8-2=6[\/latex]<\/p>\n<p>[latex]|A|=6[\/latex]\n<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gcbcecbh-R7BWKnO6xNY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/R7BWKnO6xNY?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gcbcecbh-R7BWKnO6xNY\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14660456&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-gcbcecbh-R7BWKnO6xNY&#38;vembed=0&#38;video_id=R7BWKnO6xNY&#38;video_target=tpm-plugin-gcbcecbh-R7BWKnO6xNY\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Determinant+of+a+3x3+matrix+using+Augmented+Matrices_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDeterminant of a 3x3 matrix using Augmented Matrices\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm315407\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=315407&theme=lumen&iframe_resize_id=ohm315407&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\n<div id=\"tnh-video-picker\" class=\"videoPicker\">\n<h3>Choose a Calculator<\/h3>\n<form><label>Choose Calculator:<\/label><select name=\"video\"><option value=\"\/\/plugin.3playmedia.com\/show?mf=14660457&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=Q-Dv6V2G47c&amp;video_target=tpm-plugin-drvnscs7-Q-Dv6V2G47c\">TI-84<\/option><option value=\"\/\/plugin.3playmedia.com\/show?mf=14660458&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=pXtGyKbD5uY&amp;video_target=tpm-plugin-drvnscs7-pXtGyKbD5uY\">Desmos Matrix Calculator<\/option><\/select><\/form>\n<div class=\"videoContainer threePlay\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=14660457&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=Q-Dv6V2G47c&amp;video_target=tpm-plugin-drvnscs7-Q-Dv6V2G47c\" allowfullscreen><\/iframe><\/div>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Calculating+the+Determinant+of+a+Matrix+(TI+84+Plus+CE)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCalculating the Determinant of a Matrix (TI 84 Plus CE)\u201d here (opens in new window).<\/a><br \/>\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/DESMOS+find+determinant_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDESMOS find determinant\u201d here (opens in new window).<\/a><\/section>\n<\/section>\n<h2>Use Cramer's Rule to Solve Systems in Two Variables<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Cramer's Rule:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Method using determinants to solve systems<\/li>\n<li class=\"whitespace-normal break-words\">Works when number of equations equals number of variables<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]D = 0[\/latex], system has no solution or infinite solutions<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">For system [latex]\\begin{array}{l}{a}_{1}x+{b}_{1}y={c}_{1}\\\\ {a}_{2}x+{b}_{2}y={c}_{2}\\end{array}[\/latex]:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]D = \\left\\rvert\\begin{array}{cc}{a}_{1} & {b}_{1}\\\\ {a}_{2} & {b}_{2}\\end{array}\\right\\rvert[\/latex] (coefficient matrix)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]D_x = \\left\\rvert\\begin{array}{cc}{c}_{1} & {b}_{1}\\\\ {c}_{2} & {b}_{2}\\end{array}\\right\\rvert[\/latex] (replace x-column with constants)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]D_y = \\left\\rvert\\begin{array}{cc}{a}_{1} & {c}_{1}\\\\ {a}_{2} & {c}_{2}\\end{array}\\right\\rvert[\/latex] (replace y-column with constants)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x = \\frac{D_x}{D}[\/latex], [latex]y = \\frac{D_y}{D}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Find [latex]D[\/latex] (determinant of coefficient matrix)<\/li>\n<li class=\"whitespace-normal break-words\">Find [latex]D_x[\/latex] (replace x-column with constant column)<\/li>\n<li class=\"whitespace-normal break-words\">Find [latex]D_y[\/latex] (replace y-column with constant column)<\/li>\n<li class=\"whitespace-normal break-words\">Calculate [latex]x = \\frac{D_x}{D}[\/latex] and [latex]y = \\frac{D_y}{D}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Solve using Cramer's Rule: [latex]\\begin{array}{l}12x + 3y = 15\\\\ 2x - 3y = 13\\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qcramer-001\">Show Solution<\/button><\/p>\n<div id=\"qcramer-001\" class=\"hidden-answer\" style=\"display: none\">Find [latex]D[\/latex]:[latex]D = \\left\\rvert\\begin{array}{cc}12 & 3\\\\ 2 & -3\\end{array}\\right\\rvert = 12(-3) - 2(3) = -36 - 6 = -42[\/latex]Find [latex]D_x[\/latex]:<br \/>\n[latex]D_x = \\left\\rvert\\begin{array}{cc}15 & 3\\\\ 13 & -3\\end{array}\\right\\rvert = 15(-3) - 13(3) = -45 - 39 = -84[\/latex]Find [latex]D_y[\/latex]:<br \/>\n[latex]D_y = \\left\\rvert\\begin{array}{cc}12 & 15\\\\ 2 & 13\\end{array}\\right\\rvert = 12(13) - 2(15) = 156 - 30 = 126[\/latex]Solve:<br \/>\n[latex]x = \\frac{D_x}{D} = \\frac{-84}{-42} = 2[\/latex][latex]y = \\frac{D_y}{D} = \\frac{126}{-42} = -3[\/latex]The solution is [latex](2, -3)[\/latex].\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dggdhgae-Yr9hTPvl8Ng\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Yr9hTPvl8Ng?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dggdhgae-Yr9hTPvl8Ng\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14660459&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-dggdhgae-Yr9hTPvl8Ng&#38;vembed=0&#38;video_id=Yr9hTPvl8Ng&#38;video_target=tpm-plugin-dggdhgae-Yr9hTPvl8Ng\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Learn+How+to+Use+Cramer's+Rule+to+Solve+a+2+x+2+System+of+Equations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLearn How to Use Cramer's Rule to Solve a 2 x 2 System of Equations\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm315410\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=315410&theme=lumen&iframe_resize_id=ohm315410&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Use Cramer's Rule to Solve Systems in Three Variables<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">For 3 \u00d7 3 Systems:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Same pattern as 2 \u00d7 2 systems<\/li>\n<li class=\"whitespace-normal break-words\">Replace each variable's column with constants to find that variable<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x = \\frac{D_x}{D}[\/latex], [latex]y = \\frac{D_y}{D}[\/latex], [latex]z = \\frac{D_z}{D}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Key Techniques<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Find [latex]D[\/latex] (determinant of coefficient matrix)<\/li>\n<li class=\"whitespace-normal break-words\">Find [latex]D_x[\/latex], [latex]D_y[\/latex], and [latex]D_z[\/latex] by replacing appropriate columns<\/li>\n<li class=\"whitespace-normal break-words\">Calculate each variable using the formulas<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Solve using Cramer's Rule:<br \/>\n[latex]\\begin{array}{l}x + y - z = 6\\\\ 3x - 2y + z = -5\\\\ x + 3y - 2z = 14\\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qcramer-002\">Show Solution<\/button><\/p>\n<div id=\"qcramer-002\" class=\"hidden-answer\" style=\"display: none\">Set up determinants:[latex]D = \\left\\rvert\\begin{array}{ccc}1 & 1 & -1\\\\ 3 & -2 & 1\\\\ 1 & 3 & -2\\end{array}\\right\\rvert = -3[\/latex][latex]D_x = \\left\\rvert\\begin{array}{ccc}6 & 1 & -1\\\\ -5 & -2 & 1\\\\ 14 & 3 & -2\\end{array}\\right\\rvert = -3[\/latex][latex]D_y = \\left\\rvert\\begin{array}{ccc}1 & 6 & -1\\\\ 3 & -5 & 1\\\\ 1 & 14 & -2\\end{array}\\right\\rvert = -9[\/latex][latex]D_z = \\left\\rvert\\begin{array}{ccc}1 & 1 & 6\\\\ 3 & -2 & -5\\\\ 1 & 3 & 14\\end{array}\\right\\rvert = 6[\/latex]Solve:<br \/>\n[latex]x = \\frac{-3}{-3} = 1[\/latex], [latex]y = \\frac{-9}{-3} = 3[\/latex], [latex]z = \\frac{6}{-3} = -2[\/latex]The solution is [latex](1, 3, -2)[\/latex].\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cbhdeafd-ziMrl-31PfY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/ziMrl-31PfY?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cbhdeafd-ziMrl-31PfY\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14660460&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-cbhdeafd-ziMrl-31PfY&#38;vembed=0&#38;video_id=ziMrl-31PfY&#38;video_target=tpm-plugin-cbhdeafd-ziMrl-31PfY\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+do+you+use+Cramer%E2%80%99s+Rule+to+solve+Systems+of+3+Linear+Equations%3F+The+Easiest+Method!_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow do you use Cramer\u2019s Rule to solve Systems of 3 Linear Equations? The Easiest Method!\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":25,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Finding the determinant of a 2x2 matrix | Matrices | Precalculus | Khan Academy\",\"author\":\"\",\"organization\":\"Khan Academy\",\"url\":\"https:\/\/youtu.be\/OU9sWHk_dlw\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Determinant of a 3x3 matrix using Augmented Matrices\",\"author\":\"\",\"organization\":\"East Cobb Tutoring Center\",\"url\":\"https:\/\/youtu.be\/R7BWKnO6xNY\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Calculating the Determinant of a Matrix (TI 84 Plus CE)\",\"author\":\"\",\"organization\":\"Get Your FRQ On\",\"url\":\"https:\/\/youtu.be\/Q-Dv6V2G47c\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"DESMOS find determinant\",\"author\":\"Kelly Spring\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/pXtGyKbD5uY\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Learn How to Use Cramer\\'s Rule to Solve a 2 x 2 System of Equations\",\"author\":\"\",\"organization\":\"The Math Sorcerer\",\"url\":\"https:\/\/youtu.be\/Yr9hTPvl8Ng\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"How do you use Crameru2019s Rule to solve Systems of 3 Linear Equations? 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