{"id":1381,"date":"2025-07-24T19:14:46","date_gmt":"2025-07-24T19:14:46","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1381"},"modified":"2026-03-24T19:33:03","modified_gmt":"2026-03-24T19:33:03","slug":"solving-systems-with-cramers-rule-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/solving-systems-with-cramers-rule-learn-it-3\/","title":{"raw":"Solving Systems with Cramer's Rule: Learn It 3","rendered":"Solving Systems with Cramer’s Rule: Learn It 3"},"content":{"raw":"

Using Cramer\u2019s Rule to Solve a System of Three Equations in Three Variables<\/h2>\r\nNow that we can find the determinant<\/strong> of a 3 \u00d7 3 matrix, we can apply Cramer\u2019s Rule<\/strong> to solve a system of three equations in three variables<\/strong>. Cramer\u2019s Rule is straightforward, following a pattern consistent with Cramer\u2019s Rule for 2 \u00d7 2 matrices. As the order of the matrix increases to 3 \u00d7 3, however, there are many more calculations required.\r\n\r\nWhen we calculate the determinant to be zero, Cramer\u2019s Rule gives no indication as to whether the system has no solution or an infinite number of solutions. To find out, we have to perform elimination on the system.\r\n\r\nConsider a 3 \u00d7 3 system of equations.\r\n\r\n\"A\r\n
[latex]x=\\frac{{D}_{x}}{D},y=\\frac{{D}_{y}}{D},z=\\frac{{D}_{z}}{D},D\\ne 0[\/latex]<\/div>\r\nwhere\r\n\r\n\"A\r\n\r\nIf we are writing the determinant [latex]{D}_{x}[\/latex], we replace the [latex]x[\/latex] column with the constant column. If we are writing the determinant [latex]{D}_{y}[\/latex], we replace the [latex]y[\/latex] column with the constant column. If we are writing the determinant [latex]{D}_{z}[\/latex], we replace the [latex]z[\/latex] column with the constant column. Always check the answer.\r\n\r\n
Find the solution to the given 3 \u00d7 3 system using Cramer\u2019s Rule.\r\n

[latex]\\begin{gathered}x+y-z=6\\\\ 3x - 2y+z=-5\\\\ x+3y - 2z=14\\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"180631\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"180631\"]\r\n\r\nUse Cramer\u2019s Rule.\r\n

[latex]D=\\left\\rvert\\begin{array}{ccc}1& 1& -1\\\\ 3& -2& 1\\\\ 1& 3& -2\\end{array}\\right\\rvert\\text{, }{D}_{x}=\\left\\rvert\\begin{array}{ccc}6& 1& -1\\\\ -5& -2& 1\\\\ 14& 3& -2\\end{array}\\right\\rvert\\text{, }{D}_{y}=\\left\\rvert\\begin{array}{ccc}1& 6& -1\\\\ 3& -5& 1\\\\ 1& 14& -2\\end{array}\\right\\rvert\\text{, }{D}_{z}=\\left\\rvert\\begin{array}{ccc}1& 1& 6\\\\ 3& -2& -5\\\\ 1& 3& 14\\end{array}\\right\\rvert[\/latex]<\/p>\r\nThen,\r\n

[latex]\\begin{align}x&=\\frac{{D}_{x}}{D}=\\frac{-3}{-3}=1\\hfill \\\\ y&=\\frac{{D}_{y}}{D}=\\frac{-9}{-3}=3\\hfill \\\\ z&=\\frac{{D}_{z}}{D}=\\frac{6}{-3}=-2\\hfill \\end{align}[\/latex]<\/p>\r\nThe solution is [latex]\\left(1,3,-2\\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

[ohm_question hide_question_numbers=1]321769[\/ohm_question]<\/section>
Solve the system of equations using Cramer\u2019s Rule.\r\n

[latex]\\begin{gathered}3x - 2y=4 \\\\ 6x - 4y=0\\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"966288\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"966288\"]\r\n\r\nWe begin by finding the determinants [latex]D,{D}_{x},\\text{and }{D}_{y}[\/latex].\r\n

[latex]D=\\left\\rvert\\begin{array}{cc}3& -2\\\\ 6& -4\\end{array}\\right\\rvert=3\\left(-4\\right)-6\\left(-2\\right)=0[\/latex]<\/p>\r\nWe know that a determinant of zero means that either the system has no solution or it has an infinite number of solutions. To see which one, we use the process of elimination. Our goal is to eliminate one of the variables.\r\n

    \r\n \t
  1. Multiply equation (1) by [latex]-2[\/latex].<\/li>\r\n \t
  2. Add the result to equation [latex]\\left(2\\right)[\/latex].<\/li>\r\n<\/ol>\r\n

    [latex]\\begin{align}\u22126x+4y&=\u22128 \\\\ 6x\u22124y&=0 \\\\ \\hline0&=-8\\end{align}[\/latex]<\/p>\r\nWe obtain the equation [latex]0=-8[\/latex], which is false. Therefore, the system has no solution. Graphing the system reveals two parallel lines.\r\n\r\n\"Graph\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

    Solve the system with an infinite number of solutions.\r\n

    [latex]\\begin{gathered} x - 2y+3z=0\\\\ 3x+y - 2z=0 \\\\ 2x - 4y+6z=0 \\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"282282\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"282282\"]\r\n\r\nLet\u2019s find the determinant first. Set up a matrix augmented by the first two columns.\r\n

    [latex]\\left\\rvert\\begin{array}{rrr}\\hfill 1& \\hfill -2& \\hfill 3\\\\ \\hfill 3& \\hfill 1& \\hfill -2\\\\ \\hfill 2& \\hfill -4& \\hfill 6\\end{array}\\right\\rvert\\left.\\begin{array}{rr}\\hfill 1& \\hfill -2\\\\ \\hfill 3& \\hfill 1\\\\ \\hfill 2& \\hfill -4\\end{array}\\right\\rvert[\/latex]<\/p>\r\nThen,\r\n

    [latex]1\\left(1\\right)\\left(6\\right)+\\left(-2\\right)\\left(-2\\right)\\left(2\\right)+3\\left(3\\right)\\left(-4\\right)-2\\left(1\\right)\\left(3\\right)-\\left(-4\\right)\\left(-2\\right)\\left(1\\right)-6\\left(3\\right)\\left(-2\\right)=0[\/latex]<\/p>\r\nAs the determinant equals zero, there is either no solution or an infinite number of solutions. We have to perform elimination to find out.\r\n

      \r\n \t
    1. Multiply equation (1) by [latex]-2[\/latex] and add the result to equation (3):\r\n
      [latex]\\begin{align} -2x+4y - 6z&=0\\\\ 2x - 4y+6z&=0\\\\ \\hline 0&=0 \\end{align}[\/latex]<\/div><\/li>\r\n \t
    2. Obtaining an answer of [latex]0=0[\/latex], a statement that is always true, means that the system has an infinite number of solutions. Graphing the system, we can see that two of the planes are the same and they both intersect the third plane on a line.<\/li>\r\n<\/ol>\r\n\"Two\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
      [ohm_question hide_question_numbers=1]321770[\/ohm_question]<\/section>","rendered":"

      Using Cramer\u2019s Rule to Solve a System of Three Equations in Three Variables<\/h2>\n

      Now that we can find the determinant<\/strong> of a 3 \u00d7 3 matrix, we can apply Cramer\u2019s Rule<\/strong> to solve a system of three equations in three variables<\/strong>. Cramer\u2019s Rule is straightforward, following a pattern consistent with Cramer\u2019s Rule for 2 \u00d7 2 matrices. As the order of the matrix increases to 3 \u00d7 3, however, there are many more calculations required.<\/p>\n

      When we calculate the determinant to be zero, Cramer\u2019s Rule gives no indication as to whether the system has no solution or an infinite number of solutions. To find out, we have to perform elimination on the system.<\/p>\n

      Consider a 3 \u00d7 3 system of equations.<\/p>\n

      \"A<\/p>\n

      [latex]x=\\frac{{D}_{x}}{D},y=\\frac{{D}_{y}}{D},z=\\frac{{D}_{z}}{D},D\\ne 0[\/latex]<\/div>\n

      where<\/p>\n

      \"A<\/p>\n

      If we are writing the determinant [latex]{D}_{x}[\/latex], we replace the [latex]x[\/latex] column with the constant column. If we are writing the determinant [latex]{D}_{y}[\/latex], we replace the [latex]y[\/latex] column with the constant column. If we are writing the determinant [latex]{D}_{z}[\/latex], we replace the [latex]z[\/latex] column with the constant column. Always check the answer.<\/p>\n

      Find the solution to the given 3 \u00d7 3 system using Cramer\u2019s Rule.<\/p>\n

      [latex]\\begin{gathered}x+y-z=6\\\\ 3x - 2y+z=-5\\\\ x+3y - 2z=14\\end{gathered}[\/latex]<\/p>\n