{"id":1221,"date":"2025-07-23T20:50:36","date_gmt":"2025-07-23T20:50:36","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1221"},"modified":"2026-03-22T18:07:07","modified_gmt":"2026-03-22T18:07:07","slug":"systems-of-linear-equations-in-two-variables-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/systems-of-linear-equations-in-two-variables-learn-it-5\/","title":{"raw":"Systems of Linear Equations in Two Variables: Learn It 5","rendered":"Systems of Linear Equations in 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\"A\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

[ohm_question hide_question_numbers=1]321602[\/ohm_question]<\/section>\r\n

Expressing the Solution of a System of Dependent Equations Containing Two Variables<\/h2>\r\nRecall that a dependent system<\/strong> of equations in two variables is a system in which the two equations represent the same line. Dependent systems have an infinite number of solutions because all of the points on one line are also on the other line.\r\n\r\nAfter using substitution or addition method to solve the system of equation, the resulting equation will be an identity, such as [latex]0=0[\/latex].\r\n\r\n
To write the solution of a dependent system, solve one equation for one variable, such as [latex]y = mx+b[\/latex].The\u00a0solution is often written in set notation as:\r\n

[latex](x,y) = (x, mx+b)[\/latex]<\/p>\r\nwhere [latex]x[\/latex] can be any real number.\r\n\r\n<\/section>

Solve the system of equations.\r\n

[latex]\\begin{gathered}x+3y=2\\\\ 3x+9y=6\\end{gathered}[\/latex]<\/p>\r\n\r\n[reveal-answer q=\"406019\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"406019\"]\r\nLet's use the addition method since neither equation is in the format of [latex]x=[\/latex] or [latex]y=[\/latex].\r\n\r\nLet\u2019s focus on eliminating [latex]x[\/latex]. If we multiply both sides of the first equation by [latex]-3[\/latex], then we will be able to eliminate the [latex]x[\/latex] -variable.\r\n

[latex]\\begin{align}x+3y&=2 \\\\ \\left(-3\\right)\\left(x+3y\\right)&=\\left(-3\\right)\\left(2\\right) \\\\ -3x - 9y&=-6 \\end{align}[\/latex]<\/p>\r\nNow add the equations.\r\n

[latex]\\begin{align} \u22123x\u22129y&=\u22126 \\\\ +3x+9y&=6 \\\\ \\hline 0&=0 \\end{align}[\/latex]<\/p>\r\nWe can see that there will be an infinite number of solutions<\/strong> that satisfy both equations. This is a\u00a0dependent<\/strong> system.\r\n\r\nWe can also see that this is a dependent system by graphing both equations:\r\n\r\n\"A\r\n\r\nSolution<\/strong>\r\n\r\nIf we rewrote one (or both) equations in the slope-intercept form, we might know what the solution would look like before adding. Let\u2019s look at what happens when we convert the system to slope-intercept form.\r\n

[latex]\\begin{align}\\begin{gathered}x+3y=2 \\\\ 3y=-x+2 \\\\ y=-\\frac{1}{3}x+\\frac{2}{3} \\end{gathered} \\hspace{2cm} \\begin{gathered} 3x+9y=6 \\\\9y=-3x+6 \\\\ y=-\\frac{3}{9}x+\\frac{6}{9} \\\\ y=-\\frac{1}{3}x+\\frac{2}{3} \\end{gathered}\\end{align}[\/latex]<\/p>\r\nNotice that they are the same equation of lines.\r\n\r\nThus, the general solution to the system is [latex]\\left(x, -\\frac{1}{3}x+\\frac{2}{3}\\right)[\/latex].\u00a0<\/strong>[\/hidden-answer]\r\n\r\n<\/section>

In the previous example, we presented an analysis of the solution to the following system of equations:\r\n

[latex]\\begin{gathered}x+3y=2\\\\ 3x+9y=6\\end{gathered}[\/latex]<\/p>\r\nAfter a little algebra, we found that these two equations were exactly the same. We then wrote the general solution as\u00a0[latex]\\left(x, -\\frac{1}{3}x+\\frac{2}{3}\\right)[\/latex]. Why would we write the solution this way? In some ways, this representation tells us a lot. \u00a0It tells us that [latex]x[\/latex] can be anything, [latex]x[\/latex] is [latex]x[\/latex]. \u00a0It also tells us that [latex]y[\/latex] is going to depend on [latex]x[\/latex], just like when we write a function rule. \u00a0In this case, depending on what you put in for [latex]x[\/latex], [latex]y[\/latex] will be defined in terms of [latex]x[\/latex] as [latex]-\\frac{1}{3}x+\\frac{2}{3}[\/latex].\r\n\r\nIn other words, there are infinitely many (x<\/em>,y<\/em>) pairs that will satisfy this system of equations, and they all fall on the line [latex]f(x)-\\frac{1}{3}x+\\frac{2}{3}[\/latex].\r\n\r\n<\/section>

[ohm_question hide_question_numbers=1]321606[\/ohm_question]<\/section>
[ohm_question hide_question_numbers=1]321609[\/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