{"id":1223,"date":"2025-07-23T20:48:06","date_gmt":"2025-07-23T20:48:06","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1223"},"modified":"2026-03-20T15:38:28","modified_gmt":"2026-03-20T15:38:28","slug":"systems-of-linear-equations-in-two-variables-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/systems-of-linear-equations-in-two-variables-learn-it-3\/","title":{"raw":"Systems of Linear Equations in Two Variables: Learn It 3","rendered":"Systems of Linear Equations in Two Variables: Learn It 3"},"content":{"raw":"

Solving Systems of Equations by Substitution<\/h2>\r\nSolving a linear system in two variables by graphing works well when the solution consists of integer values, but if our solution contains decimals or fractions, it is not the most precise method. We will consider two more methods of solving a system of linear equations that are more precise than graphing.\r\n\r\nOne such method is solving a system of equations by the substitution method<\/strong>, in which we solve one of the equations for one variable and then substitute the result into the second equation to solve for the second variable. Recall that we can solve for only one variable at a time, which is the reason the substitution method is both valuable and practical.\r\n\r\n
How To: Given a system of two equations in two variables, solve using the substitution method.<\/strong>\r\n
    \r\n \t
  1. Solve one of the two equations for one of the variables in terms of the other.<\/li>\r\n \t
  2. Substitute the expression for this variable into the second equation, then solve for the remaining variable.<\/li>\r\n \t
  3. Substitute that solution into either of the original equations to find the value of the first variable. If possible, write the solution as an ordered pair.<\/li>\r\n \t
  4. Check the solution in both equations.<\/li>\r\n<\/ol>\r\n<\/section>
    Solve the following system of equations by substitution.\r\n

    [latex]\\begin{align}-x+y&=-5 \\\\ 2x-5y&=1 \\end{align}[\/latex]<\/p>\r\n[reveal-answer q=\"786744\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"786744\"]\r\n\r\nFirst, we will solve the first equation for [latex]y[\/latex].\r\n

    [latex]\\begin{align}-x+y&=-5 \\\\ y&=x - 5 \\end{align}[\/latex]<\/p>\r\nNow we can substitute the expression [latex]x - 5[\/latex] for [latex]y[\/latex] in the second equation.\r\n

    [latex]\\begin{align}2x - 5y&=1 \\\\ 2x - 5\\left(x - 5\\right)&=1 \\\\ 2x - 5x+25&=1 \\\\ -3x&=-24 \\\\ x&=8 \\end{align}[\/latex]<\/p>\r\nNow, we substitute [latex]x=8[\/latex] into the first equation and solve for [latex]y[\/latex].\r\n

    [latex]\\begin{align}-\\left(8\\right)+y&=-5 \\\\ y&=3 \\end{align}[\/latex]<\/p>\r\nOur solution is [latex]\\left(8,3\\right)[\/latex].\r\n\r\nCheck the solution by substituting [latex]\\left(8,3\\right)[\/latex] into both equations.\r\n

    [latex]\\begin{align}-x+y&=-5 \\\\ -\\left(8\\right)+\\left(3\\right)&=-5 && \\text{True} \\\\[3mm] 2x - 5y&=1 \\\\ 2\\left(8\\right)-5\\left(3\\right)&=1 && \\text{True} \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

    Solve the following system of equations.\r\n

    [latex]\\begin{gathered}&x=9 - 2y \\\\ &x+2y=13 \\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"888134\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"888134\"]\r\n\r\nBecause one equation is already solved for [latex]x[\/latex], we can substitute\u00a0 [latex]x=9 - 2y[\/latex] into the second equation:\r\n

    [latex]\\begin{align}x+2y&=13 \\\\ \\left(9 - 2y\\right)+2y&=13 \\\\ 9+0y&=13 \\\\ 9&=13 \\end{align}[\/latex]<\/p>\r\nClearly, this statement is a contradiction because [latex]9\\ne 13[\/latex]. Therefore, the system has no solution<\/strong>.\r\n\r\n\"A\r\n\r\nAnalysis of the Solution<\/strong>\r\n\r\nLet's graph the equations to confirm that the system has no solution. Writing the equations in slope-intercept form confirms that the system is inconsistent<\/strong> because all lines will intersect eventually unless they are parallel. Parallel lines will never intersect; thus, the two lines have no points in common.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

    [ohm_question hide_question_numbers=1]321588[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]321589[\/ohm_question]<\/section>","rendered":"

    Solving Systems of Equations by Substitution<\/h2>\n

    Solving a linear system in two variables by graphing works well when the solution consists of integer values, but if our solution contains decimals or fractions, it is not the most precise method. We will consider two more methods of solving a system of linear equations that are more precise than graphing.<\/p>\n

    One such method is solving a system of equations by the substitution method<\/strong>, in which we solve one of the equations for one variable and then substitute the result into the second equation to solve for the second variable. Recall that we can solve for only one variable at a time, which is the reason the substitution method is both valuable and practical.<\/p>\n

    How To: Given a system of two equations in two variables, solve using the substitution method.<\/strong><\/p>\n
      \n
    1. Solve one of the two equations for one of the variables in terms of the other.<\/li>\n
    2. Substitute the expression for this variable into the second equation, then solve for the remaining variable.<\/li>\n
    3. Substitute that solution into either of the original equations to find the value of the first variable. If possible, write the solution as an ordered pair.<\/li>\n
    4. Check the solution in both equations.<\/li>\n<\/ol>\n<\/section>\n
      Solve the following system of equations by substitution.<\/p>\n

      [latex]\\begin{align}-x+y&=-5 \\\\ 2x-5y&=1 \\end{align}[\/latex]<\/p>\n