{"id":2327,"date":"2024-07-22T22:23:09","date_gmt":"2024-07-22T22:23:09","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=2327"},"modified":"2025-08-15T15:38:27","modified_gmt":"2025-08-15T15:38:27","slug":"systems-of-linear-equations-two-variables-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/systems-of-linear-equations-two-variables-learn-it-5\/","title":{"raw":"Systems of Linear Equations Two Variables: Learn It 5","rendered":"Systems of Linear Equations Two Variables: Learn It 5"},"content":{"raw":"

Identifying Inconsistent Systems of Equations Containing Two Variables<\/h2>\r\nNow that we have several methods for solving systems of equations, we can use the methods to identify inconsistent systems. Recall that an inconsistent system<\/strong> consists of parallel lines that have the same slope but different [latex]y[\/latex] -intercepts. They will never intersect. When searching for a solution to an inconsistent system, we will come up with a false statement, such as [latex]12=0[\/latex].\r\n\r\n
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\nWe can approach this problem in two ways. Because one equation is already solved for [latex]x[\/latex], the most obvious step is to use substitution.\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.\r\n\r\nThe second approach would be to first manipulate the equations so that they are both in slope-intercept form. We manipulate the first equation as follows.\r\n

[latex]\\begin{gathered}x=9 - 2y \\\\ 2y=-x+9 \\\\ y=-\\frac{1}{2}x+\\frac{9}{2} \\end{gathered}[\/latex]<\/p>\r\nWe then convert the second equation expressed to slope-intercept form.\r\n

[latex]\\begin{gathered}x+2y=13 \\\\ 2y=-x+13 \\\\ y=-\\frac{1}{2}x+\\frac{13}{2} \\end{gathered}[\/latex]<\/p>\r\nComparing the equations, we see that they have the same slope but different y<\/em>-intercepts. Therefore, the lines are parallel and do not intersect.\r\n

[latex]\\begin{gathered}y=-\\frac{1}{2}x+\\frac{9}{2} \\\\ y=-\\frac{1}{2}x+\\frac{13}{2} \\end{gathered}[\/latex]<\/p>\r\nAnalysis of the Solution<\/strong>\r\n\r\nWriting the equations in slope-intercept form confirms that the system is inconsistent because all lines will intersect eventually unless they are parallel. Parallel lines will never intersect; thus, the two lines have no points in common. The graphs of the equations in this example are shown below.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]\"A A graph of two parallel lines[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

[ohm_question hide_question_numbers=1]291790[\/ohm_question]<\/section> ","rendered":"

Identifying Inconsistent Systems of Equations Containing Two Variables<\/h2>\n

Now that we have several methods for solving systems of equations, we can use the methods to identify inconsistent systems. Recall that an inconsistent system<\/strong> consists of parallel lines that have the same slope but different [latex]y[\/latex] -intercepts. They will never intersect. When searching for a solution to an inconsistent system, we will come up with a false statement, such as [latex]12=0[\/latex].<\/p>\n

Solve the following system of equations.<\/p>\n

[latex]\\begin{gathered}&x=9 - 2y \\\\ &x+2y=13 \\end{gathered}[\/latex]<\/p>\n