{"id":2914,"date":"2024-02-09T20:20:13","date_gmt":"2024-02-09T20:20:13","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&p=2914"},"modified":"2026-03-11T15:50:37","modified_gmt":"2026-03-11T15:50:37","slug":"8-3-2-the-elimination-method-ii","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/8-3-2-the-elimination-method-ii\/","title":{"raw":"8.3.2: The Elimination Method II","rendered":"8.3.2: The Elimination Method II"},"content":{"raw":"
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Learning Objectives<\/h1>\r\n
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  • Solve a system of linear equation using the elimination method<\/li>\r\n<\/ul>\r\n<\/div>\r\n

    The Elimination Method<\/h2>\r\nIn the last section, we solved systems of linear equations using the\u00a0elimination method<\/strong><\/em>. In this method, we add equations where one of the variables has opposite coefficients in the equations, so that the sum is zero. This eliminates that variable and results in a linear equation in one variable which can be solved, a contradiction that implies no solution, or an identity which implies infinite solutions constrained by either equation. Of course, not all systems are set up with the two terms of one variable having opposite coefficients. Often we must adjust one or both of the equations by multiplication so that one variable will be eliminated by addition.\r\n
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    the ELIMINATION method<\/h3>\r\n
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    1. Write both equations with [latex]x[\/latex]\u2013 and\u00a0[latex]y[\/latex]-variables on the left side of the equals sign and constants on the right.<\/li>\r\n \t
    2. Write one equation above the other, lining up corresponding variables. If one of the variables in the top equation has the opposite coefficient of the same variable in the bottom equation, add the equations together, eliminating one variable. If not, use multiplication by nonzero numbers so that one of the variables in the top equation has the opposite coefficient of the same variable in the bottom equation, then add the equations to eliminate the variable.<\/li>\r\n \t
    3. Solve the resulting equation for the remaining variable. If a contradiction occurs, there is no solution. If an identity occurs, there are infinite solutions constrained by either equation.<\/li>\r\n \t
    4. Substitute that value into one of the original equations and solve for the second variable.<\/li>\r\n \t
    5. Check the solution by substituting the values into the other equation.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
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      Example<\/h3>\r\nSolve the system of equations:\r\n

      [latex]\\begin{align}x+2y&=-1 \\\\ -x+\\;y&=3 \\end{align}[\/latex]<\/p>\r\n\r\n

      Solution<\/h4>\r\nBoth equations are already set equal to a constant. Notice that the coefficient of [latex]x[\/latex] in the second equation, \u20131, is the opposite of the coefficient of [latex]x[\/latex] in the first equation, 1. We can add the two equations to eliminate [latex]x[\/latex] without needing to multiply by a constant:\r\n

      [latex]\\begin{align} x+2y&=-1 \\\\ -x+y&=3 \\\\ \\hline 0x+3y&=2\\end{align}[\/latex]<\/p>\r\nNow that we have eliminated [latex]x[\/latex], we can solve the resulting equation for [latex]y[\/latex]:\r\n

      [latex]\\begin{align}3y&=2 \\\\ y&=\\dfrac{2}{3} \\end{align}[\/latex]<\/p>\r\nNow substitute this value for [latex]y[\/latex] into one of the original equations and solve for [latex]x[\/latex]:\r\n

      [latex]\\begin{align}-x+y&=3 \\\\ -x+\\frac{2}{3}&=3 \\\\ -x&=3-\\frac{2}{3} \\\\ -x&=\\frac{7}{3} \\\\ x&=-\\frac{7}{3} \\end{align}[\/latex]<\/p>\r\n

      The solution to this system is [latex]\\left(-\\frac{7}{3},\\frac{2}{3}\\right)[\/latex].<\/p>\r\nCheck the solution in the other equation:\r\n

      [latex]\\begin{align}x+2y&=-1 \\\\ \\left(-\\frac{7}{3}\\right)+2\\left(\\frac{2}{3}\\right)&= \\\\ -\\frac{7}{3}+\\frac{4}{3}&= \\\\ -\\frac{3}{3}&= \\\\ -1&=-1&& \\text{True} \\end{align}[\/latex]<\/p>\r\n\r\n

      Solution<\/h4>\r\nThe solution is\u00a0[latex]\\left(-\\frac{7}{3},\\frac{2}{3}\\right)[\/latex].\r\n

      Analysis of the Solution<\/h4>\r\nWe gain an important perspective on systems of equations by looking at the graphical representation. See the graph below\u00a0to find that the equations intersect at the solution. We do not need to ask whether there may be a second solution because observing the graph confirms that the system has exactly one solution.\r\n
      \"Two<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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      Try It<\/h3>\r\n