{"id":2907,"date":"2024-02-09T20:17:45","date_gmt":"2024-02-09T20:17:45","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&p=2907"},"modified":"2026-03-11T14:44:04","modified_gmt":"2026-03-11T14:44:04","slug":"8-1-2-solve-a-system-of-linear-equations-by-graphing","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/8-1-2-solve-a-system-of-linear-equations-by-graphing\/","title":{"raw":"8.1.2 Solve a System of Linear Equations by Graphing","rendered":"8.1.2 Solve a System of Linear Equations by Graphing"},"content":{"raw":"
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Learning Outcomes<\/h1>\r\n
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  • Graph a system of linear equations in two variables.<\/li>\r\n \t
  • Determine the solution of a system of equations graphically.<\/li>\r\n \t
  • Determine the type of system from the graph.<\/li>\r\n<\/ul>\r\n<\/div>\r\n
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    Key words<\/h1>\r\n
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    • Parallel lines<\/strong>: lines that have the same slope but different [latex]y[\/latex]-intercepts<\/li>\r\n \t
    • Coinciding lines<\/strong>: lines that have the same slope and the same[latex]y[\/latex]-intercept<\/li>\r\n<\/ul>\r\n<\/div>\r\n

      Graph a system of linear equations<\/h2>\r\nThere are multiple methods for solving systems of linear equations. For a\u00a0system of linear equations<\/em>\u00a0<\/strong>in two variables, we can visually determine both the type of system and the solution by graphing the system of equations on the same set of axes.\r\n\r\nWe will practice graphing two equations on the same set of axes,\u00a0and explore the considerations required when graphing two linear inequalities on the same set of axes. The same techniques are used to graph a system of linear equations as as we used to graph single linear equations. We can use tables of values, slope and [latex]y[\/latex]-intercept, or [latex]x[\/latex]\u2013 and [latex]y[\/latex]-intercepts to graph each line on the same set of axes.\r\n
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      Example<\/h3>\r\nSolve the following system of equations by graphing. Identify the type of system.\r\n

      [latex]\\begin{array}{c}2x+y=-8\\\\ x-y=-1\\end{array}[\/latex]<\/p>\r\n\r\n

      Solution<\/h4>\r\nTo graph an equation in the form [latex]ax+by=c[\/latex], we can find the intercepts.\r\n\r\n \r\n\r\nEquation 1:\u00a0 \u00a0 [latex]2x+y=-8[\/latex]\r\n\r\nSet [latex]x=0[\/latex] then [latex]y=-8[\/latex]. [latex]\\left (0, -8\\right )[\/latex] is the [latex]y[\/latex]-intercept.\r\n\r\nSet\u00a0[latex]y=0[\/latex] then [latex]2x=-8[\/latex], so [latex]x=-4[\/latex]. [latex]\\left (-4, 0\\right )[\/latex] is the [latex]x[\/latex]-intercept.\r\n\r\n \r\n\r\nEquation 2:\u00a0 \u00a0 [latex]x-y=-1[\/latex]\r\n\r\nSet [latex]x=0[\/latex] then [latex]-y=-1[\/latex], so [latex]y=1[\/latex].\u00a0 [latex]\\left (0, 1\\right )[\/latex] is the [latex]y[\/latex]-intercept.\r\n\r\nSet\u00a0[latex]y=0[\/latex] then [latex]x=-1[\/latex].\u00a0 [latex]\\left (-4, 0\\right )[\/latex] is the [latex]x[\/latex]-intercept.\r\n\r\nGraph both equations on the same set of axes.\r\n
      \"The<\/div>\r\nThe lines appear to intersect at the point [latex]\\left(-3,-2\\right)[\/latex]. We can check to make sure that this is the solution to the system by substituting the ordered pair into both equations\r\n

      [latex]\\begin{array}{ll}2\\left(-3\\right)+\\left(-2\\right)=-8\\hfill & \\hfill \\\\ \\text{ }-8=-8\\hfill & \\text{True}\\hfill \\\\ \\text{ }\\left(-3\\right)-\\left(-2\\right)=-1\\hfill & \\hfill \\\\ \\text{ }-1=-1\\hfill & \\text{True}\\hfill \\end{array}[\/latex]<\/p>\r\nThe solution to the system is the ordered pair [latex]\\left(-3,-2\\right)[\/latex].\r\n\r\nThe system is independent since we have two lines crossing at a single point.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n \r\n\r\nWatch the video below for another example of how to solve a system of equations by first graphing the lines and then identifying the solution the system has.\r\n\r\n