2.7.2 Systems of Linear Equations with No Solution

Solving an Inconsistent System of Equations

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 consists of parallel lines that have the same slope but different $y$-intercepts. They will never intersect. When searching for a solution to an inconsistent system, we will come up with a false statement, such as $12=0$.

Solve the following system of equations.

$x=9-2y$

$x+2y=13$

Solution

We can approach this problem in two ways. Because one equation is already solved for $x$, the most obvious step is to use substitution.

$\begin{array}{ccc}\hfill x+2y& \hfill =\hfill & 13\hfill \\ \hfill (9-2y)+2y& \hfill =\hfill & 13\hfill \\ \hfill 9+0y& \hfill =\hfill & 13\hfill \\ \hfill 9& \hfill =\hfill & 13\hfill \end{array}$

Clearly, this statement is a contradiction because $9\ne 13$. Therefore, the system has no solution.

The 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:

$x=9-2y$

$2y=-x+9x$

$y=-\frac{1}{2}x+\frac{9}{2}$

We then convert the second equation expressed to slope-intercept form.

$\begin{array}{ccc}\hfill x+2y& \hfill =\hfill & 13\hfill \\ \hfill 2y& \hfill =\hfill & -x+13\hfill \\ \hfill y& \hfill =\hfill & -\frac{1}{2}x+\frac{13}{2}\hfill \end{array}$

Comparing the equations, we see that they have the same slope but different $y$-intercepts. Therefore, the lines are parallel and do not intersect.

$y=-\frac{1}{2}x+\frac{9}{2}$

$y=-\frac{1}{2}x+\frac{13}{2}$