{"id":2905,"date":"2024-02-09T20:17:05","date_gmt":"2024-02-09T20:17:05","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&p=2905"},"modified":"2026-03-11T14:30:24","modified_gmt":"2026-03-11T14:30:24","slug":"8-1-1-systems-of-linear-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/8-1-1-systems-of-linear-equations\/","title":{"raw":"8.1.1 Systems of Linear Equations","rendered":"8.1.1 Systems of Linear Equations"},"content":{"raw":"
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Learning Objectives<\/h1>\r\n
    \r\n \t
  • Determine if an ordered pair is a solution of the system of equations.<\/li>\r\n \t
  • Classify a system of equations as consistent or inconsistent.<\/li>\r\n \t
  • Classify a system of consistent equations as independent or dependent.<\/li>\r\n<\/ul>\r\n<\/div>\r\n
    \r\n

    Key words<\/h1>\r\n
      \r\n \t
    • system of linear equations<\/strong>:\u00a0two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously<\/li>\r\n \t
    • parallel lines<\/strong>: lines with the same slope but different\u00a0[latex]y[\/latex]-intercepts<\/li>\r\n \t
    • coinciding lines<\/strong>: lines with the same slope and\u00a0the same [latex]y[\/latex]-intercept<\/li>\r\n \t
    • consistent system<\/strong>: a system of equations that has at least one solution (the lines represented by the equations are not parallel)<\/li>\r\n \t
    • inconsistent system<\/strong>:\u00a0a system of equations that no solution (the lines represented by the equations are parallel)<\/li>\r\n \t
    • independent system<\/strong>: <\/strong>the lines represented by each equation have different slopes, or have the same slope but different [latex]y[\/latex]-intercepts (the lines represented by the equations either cross at one point or are parallel)<\/li>\r\n \t
    • dependent system<\/strong>:\u00a0<\/strong>the lines represented by each equation have the same slope and the same [latex]y[\/latex]-intercept (the lines represented by the equations are coincidental)<\/li>\r\n<\/ul>\r\n<\/div>\r\n

      Systems of Linear Equations<\/h2>\r\n
      \r\n
      \r\n\r\nA skateboard manufacturer introduces a new line of boards. The manufacturer tracks its costs, which is the amount it spends to produce the boards, and its revenue, which is the amount it earns through sales of its boards. How can the company determine if it is making a profit with its new line? How many skateboards must be produced and sold before a profit is possible? In this section, we will consider linear equations with two variables to answer these and similar questions.\r\n

      Solutions<\/h2>\r\nIn order to investigate situations such as that of the skateboard manufacturer, we need to recognize that we are dealing with more than one variable and likely more than one equation. A\u00a0system of linear equations\u00a0<\/em><\/strong>consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all<\/span> equations in the system at the same time. Some linear systems may not have a solution and others may have an infinite number of solutions. In order for a linear system to have a unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a unique solution.\r\n\r\nIn this section, we will look at systems of linear equations in two variables, which consist of two linear equations that contain two variables. For example, consider the following system of linear equations in two variables.\r\n

      [latex]\\begin{array}{c}2x+y=\\text{ }15\\\\ 3x-y=\\text{ }5\\end{array}[\/latex]<\/p>\r\nThe\u00a0solution\u00a0<\/strong><\/em>to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. In this example, the ordered pair (4, 7) is the solution to the system of linear equations. We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations. Shortly we will investigate methods of finding such a solution if it exists.\r\n

      [latex]\\begin{array}{l}2\\left(4\\right)+\\left(7\\right)=15\\text{ }\\text{True}\\hfill \\\\ 3\\left(4\\right)-\\left(7\\right)=5\\text{ }\\text{True}\\hfill \\end{array}[\/latex]<\/p>\r\nA system of two linear equations can have one solution, an infinite number of solutions, or no solution. To see this visually, consider that each of the linear equations can be represented by a line, and there is only three possibilities with two lines:\r\n\r\n[caption id=\"attachment_1983\" align=\"alignleft\" width=\"254\"]\"crossing Figure 1. Lines cross in a single point[\/caption]\r\n\r\n[caption id=\"attachment_1984\" align=\"alignleft\" width=\"228\"]\"Parallel Figure 2. Lines are parallel[\/caption]\r\n\r\n[caption id=\"attachment_1985\" align=\"alignleft\" width=\"213\"]\"coincidental Figure 3. Lines are coinciding[\/caption]\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n<\/div>\r\n

      \r\n\r\nFor a system of equations to have a unique solution, they must cross at a single point. Parallel lines never cross, so the system has no solution. Coinciding lines lie on top of one another so cross at all points on the line resulting in infinite solutions.\r\n\r\nSystems of equations can be classified by the number of solutions. A\u00a0consistent system\u00a0<\/em><\/strong>of equations has at least one solution. A consistent system is considered to be an\u00a0independent system\u00a0<\/em><\/strong>if it has a single solution. This means that the two lines have different slopes and intersect at one point in the plane.\r\n\r\nA consistent system is considered to be a\u00a0dependent system\u00a0<\/em><\/strong>if the equations have the same slope and the same\r\ny<\/em>-intercept. In other words, the lines coincide so the equations represent the same line. Every point on the line represents a coordinate pair that satisfies the system. Thus, there are an infinite number of solutions. lying on the line.\r\n\r\nAnother type of system of linear equations is an\u00a0inconsistent system<\/strong><\/em>, which is one in which the equations represent two parallel lines. The lines have the same slope and different\u00a0y-<\/em>intercepts. There are no points common to both lines; hence, there is no solution to the system.\r\n
      \r\n

      \u00a0Types of Linear Systems<\/h3>\r\nThere are three types of systems of linear equations in two variables, and three types of solutions.\r\n
        \r\n \t
      • An\u00a0independent system\u00a0<\/strong>has exactly one solution pair [latex]\\left(x,y\\right)[\/latex]. The point where the two lines intersect is the only solution.<\/li>\r\n \t
      • An\u00a0inconsistent system\u00a0<\/strong>has no solution. Notice that the two lines are parallel and will never intersect.<\/li>\r\n \t
      • A\u00a0dependent system\u00a0<\/strong>has infinitely many solutions. The lines are coinciding. They are the same line, so every coordinate pair on the line is a solution to both equations.<\/li>\r\n<\/ul>\r\nBelow is a comparison of\u00a0graphical representations of each type of system.\r\n
        \"Independent<\/div>\r\n<\/div>\r\n
        \r\n

        \u00a0solution of a system of equations<\/h3>\r\n
          \r\n \t
        1. Substitute the ordered pair into each equation in the system.<\/li>\r\n \t
        2. Determine whether true statements result from the substitution in both equations; if so, the ordered pair is a solution.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
          \r\n

          Example<\/h3>\r\nDetermine whether the ordered pair [latex]\\left(5,1\\right)[\/latex] is a solution to the given system of equations.\r\n

          [latex]\\begin{array}{l}x+3y=8\\hfill \\\\ 2x - 9=y\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n

          Solution<\/h4>\r\nSubstitute the ordered pair [latex]\\left(5,1\\right)[\/latex] into both equations.\r\n

          [latex]\\begin{array}{ll}\\left(5\\right)+3\\left(1\\right)=8\\hfill & \\hfill \\\\ \\text{ }8=8\\hfill & \\text{True}\\hfill \\\\ 2\\left(5\\right)-9=\\left(1\\right)\\hfill & \\hfill \\\\ \\text{ }\\text{1=1}\\hfill & \\text{True}\\hfill \\end{array}[\/latex]<\/p>\r\nThe ordered pair [latex]\\left(5,1\\right)[\/latex] satisfies both equations, so it is the solution to the system.\r\n

          Analysis of the Solution<\/h4>\r\nWe can see the solution clearly by plotting the graph of each equation. Since the solution is an ordered pair that satisfies both equations, it is a point on both of the lines and thus the point of intersection of the two lines.\r\n\r\n\"Two\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
          \r\n

          Try It<\/h3>\r\nDetermine whether the ordered pair [latex]\\left(8,5\\right)[\/latex] is a solution to the following system.\r\n

          [latex]\\begin{array}{c}5x - 4y=20\\\\ 2x+1=3y\\end{array}[\/latex]<\/p>\r\n[reveal-answer q=\"hjm643\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm643\"]\r\n\r\nTo be a solution of the system, it must be a solution to each equation:\r\n\r\n[latex]5x - 4y=20 \\\\ 5(8)-4(5)=20 \\\\ 40-20=20 \\\\\\text{checks}[\/latex]\r\n\r\n[latex]2x +1=3y \\\\ 2(8)+1=3(5) \\\\ 16+1=15 \\\\\\text{does not check}[\/latex]\r\n\r\n[latex]\\left(8,5\\right)[\/latex] is not a solution.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n ","rendered":"

          \n
          \n
          \n
          \n

          Learning Objectives<\/h1>\n
            \n
          • Determine if an ordered pair is a solution of the system of equations.<\/li>\n
          • Classify a system of equations as consistent or inconsistent.<\/li>\n
          • Classify a system of consistent equations as independent or dependent.<\/li>\n<\/ul>\n<\/div>\n
            \n

            Key words<\/h1>\n
              \n
            • system of linear equations<\/strong>:\u00a0two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously<\/li>\n
            • parallel lines<\/strong>: lines with the same slope but different\u00a0[latex]y[\/latex]-intercepts<\/li>\n
            • coinciding lines<\/strong>: lines with the same slope and\u00a0the same [latex]y[\/latex]-intercept<\/li>\n
            • consistent system<\/strong>: a system of equations that has at least one solution (the lines represented by the equations are not parallel)<\/li>\n
            • inconsistent system<\/strong>:\u00a0a system of equations that no solution (the lines represented by the equations are parallel)<\/li>\n
            • independent system<\/strong>: <\/strong>the lines represented by each equation have different slopes, or have the same slope but different [latex]y[\/latex]-intercepts (the lines represented by the equations either cross at one point or are parallel)<\/li>\n
            • dependent system<\/strong>:\u00a0<\/strong>the lines represented by each equation have the same slope and the same [latex]y[\/latex]-intercept (the lines represented by the equations are coincidental)<\/li>\n<\/ul>\n<\/div>\n

              Systems of Linear Equations<\/h2>\n
              \n
              \n

              A skateboard manufacturer introduces a new line of boards. The manufacturer tracks its costs, which is the amount it spends to produce the boards, and its revenue, which is the amount it earns through sales of its boards. How can the company determine if it is making a profit with its new line? How many skateboards must be produced and sold before a profit is possible? In this section, we will consider linear equations with two variables to answer these and similar questions.<\/p>\n

              Solutions<\/h2>\n

              In order to investigate situations such as that of the skateboard manufacturer, we need to recognize that we are dealing with more than one variable and likely more than one equation. A\u00a0system of linear equations\u00a0<\/em><\/strong>consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all<\/span> equations in the system at the same time. Some linear systems may not have a solution and others may have an infinite number of solutions. In order for a linear system to have a unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a unique solution.<\/p>\n

              In this section, we will look at systems of linear equations in two variables, which consist of two linear equations that contain two variables. For example, consider the following system of linear equations in two variables.<\/p>\n

              [latex]\\begin{array}{c}2x+y=\\text{ }15\\\\ 3x-y=\\text{ }5\\end{array}[\/latex]<\/p>\n

              The\u00a0solution\u00a0<\/strong><\/em>to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. In this example, the ordered pair (4, 7) is the solution to the system of linear equations. We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations. Shortly we will investigate methods of finding such a solution if it exists.<\/p>\n

              [latex]\\begin{array}{l}2\\left(4\\right)+\\left(7\\right)=15\\text{ }\\text{True}\\hfill \\\\ 3\\left(4\\right)-\\left(7\\right)=5\\text{ }\\text{True}\\hfill \\end{array}[\/latex]<\/p>\n

              A system of two linear equations can have one solution, an infinite number of solutions, or no solution. To see this visually, consider that each of the linear equations can be represented by a line, and there is only three possibilities with two lines:<\/p>\n

              \"crossing<\/p>\n

              Figure 1. Lines cross in a single point<\/p>\n<\/div>\n

              \"Parallel<\/p>\n

              Figure 2. Lines are parallel<\/p>\n<\/div>\n

              \"coincidental<\/p>\n

              Figure 3. Lines are coinciding<\/p>\n<\/div>\n

               <\/p>\n

               <\/p>\n

               <\/p>\n

               <\/p>\n

               <\/p>\n

               <\/p>\n<\/div>\n

              \n

              For a system of equations to have a unique solution, they must cross at a single point. Parallel lines never cross, so the system has no solution. Coinciding lines lie on top of one another so cross at all points on the line resulting in infinite solutions.<\/p>\n

              Systems of equations can be classified by the number of solutions. A\u00a0consistent system\u00a0<\/em><\/strong>of equations has at least one solution. A consistent system is considered to be an\u00a0independent system\u00a0<\/em><\/strong>if it has a single solution. This means that the two lines have different slopes and intersect at one point in the plane.<\/p>\n

              A consistent system is considered to be a\u00a0dependent system\u00a0<\/em><\/strong>if the equations have the same slope and the same
              \ny<\/em>-intercept. In other words, the lines coincide so the equations represent the same line. Every point on the line represents a coordinate pair that satisfies the system. Thus, there are an infinite number of solutions. lying on the line.<\/p>\n

              Another type of system of linear equations is an\u00a0inconsistent system<\/strong><\/em>, which is one in which the equations represent two parallel lines. The lines have the same slope and different\u00a0y-<\/em>intercepts. There are no points common to both lines; hence, there is no solution to the system.<\/p>\n

              \n

              \u00a0Types of Linear Systems<\/h3>\n

              There are three types of systems of linear equations in two variables, and three types of solutions.<\/p>\n

                \n
              • An\u00a0independent system\u00a0<\/strong>has exactly one solution pair [latex]\\left(x,y\\right)[\/latex]. The point where the two lines intersect is the only solution.<\/li>\n
              • An\u00a0inconsistent system\u00a0<\/strong>has no solution. Notice that the two lines are parallel and will never intersect.<\/li>\n
              • A\u00a0dependent system\u00a0<\/strong>has infinitely many solutions. The lines are coinciding. They are the same line, so every coordinate pair on the line is a solution to both equations.<\/li>\n<\/ul>\n

                Below is a comparison of\u00a0graphical representations of each type of system.<\/p>\n

                \"Independent<\/div>\n<\/div>\n
                \n

                \u00a0solution of a system of equations<\/h3>\n
                  \n
                1. Substitute the ordered pair into each equation in the system.<\/li>\n
                2. Determine whether true statements result from the substitution in both equations; if so, the ordered pair is a solution.<\/li>\n<\/ol>\n<\/div>\n
                  \n

                  Example<\/h3>\n

                  Determine whether the ordered pair [latex]\\left(5,1\\right)[\/latex] is a solution to the given system of equations.<\/p>\n

                  [latex]\\begin{array}{l}x+3y=8\\hfill \\\\ 2x - 9=y\\hfill \\end{array}[\/latex]<\/p>\n

                  Solution<\/h4>\n

                  Substitute the ordered pair [latex]\\left(5,1\\right)[\/latex] into both equations.<\/p>\n

                  [latex]\\begin{array}{ll}\\left(5\\right)+3\\left(1\\right)=8\\hfill & \\hfill \\\\ \\text{ }8=8\\hfill & \\text{True}\\hfill \\\\ 2\\left(5\\right)-9=\\left(1\\right)\\hfill & \\hfill \\\\ \\text{ }\\text{1=1}\\hfill & \\text{True}\\hfill \\end{array}[\/latex]<\/p>\n

                  The ordered pair [latex]\\left(5,1\\right)[\/latex] satisfies both equations, so it is the solution to the system.<\/p>\n

                  Analysis of the Solution<\/h4>\n

                  We can see the solution clearly by plotting the graph of each equation. Since the solution is an ordered pair that satisfies both equations, it is a point on both of the lines and thus the point of intersection of the two lines.<\/p>\n

                  \"Two<\/p>\n<\/div>\n<\/div>\n<\/div>\n

                  \n

                  Try It<\/h3>\n

                  Determine whether the ordered pair [latex]\\left(8,5\\right)[\/latex] is a solution to the following system.<\/p>\n

                  [latex]\\begin{array}{c}5x - 4y=20\\\\ 2x+1=3y\\end{array}[\/latex]<\/p>\n

                  Show Answer<\/span><\/p>\n
                  \n

                  To be a solution of the system, it must be a solution to each equation:<\/p>\n

                  [latex]5x - 4y=20 \\\\ 5(8)-4(5)=20 \\\\ 40-20=20 \\\\\\text{checks}[\/latex]<\/p>\n

                  [latex]2x +1=3y \\\\ 2(8)+1=3(5) \\\\ 16+1=15 \\\\\\text{does not check}[\/latex]<\/p>\n

                  [latex]\\left(8,5\\right)[\/latex] is not a solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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