4.2 Introduction to systems of linear equations: solving by graphing

The lines intersect at the point $\left(a,b\right)$ . The point $\left(a,b\right)$ is the solution to the corresponding system.

The lines are parallel. They do not intersect. The system has no solution.

The lines are coincident (one on the other). They intersect at infinitely many points. The system has infinitely many solutions.

$\{\begin{array}{ccc}\begin{array}{r}2x+y=5\\ x+y=2\end{array}& & \begin{array}{l}\left(1\right)\\ \left(2\right)\end{array}\end{array}$
Write each equation in slope-intercept form.

$\begin{array}{lllllll}\left(1\right)\hfill & \hfill & -2x+y=5\hfill & \hfill & \left(2\right)\hfill & \hfill & x+y=2\hfill \\ \hfill & \hfill & y=2x+5\hfill & \hfill & \hfill & \hfill & y=-x+2\hfill \end{array}$
Graph each of these equations.

The lines appear to intersect at the point $\left(-1,3\right)$ . The solution to this system is $\left(-1,3\right)$ , or

$\begin{array}{ccc}x=-1,& & y=3\end{array}$

Check:   Substitute $x=-1,y=3$ into each equation.

$\begin{array}{llllllllllll}(1)\hfill & \hfill & \hfill -2x+y& =\hfill & 5\hfill & \hfill & (2)\hfill & \hfill & \hfill x+y& =\hfill & 2\hfill & \hfill \\ \hfill & \hfill & \hfill -2(-1)+3& =\hfill & 5\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & \hfill & \hfill & \hfill -1+3& =\hfill & 2\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 2+3& =\hfill & 5\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & \hfill & \hfill & \hfill 2& =\hfill & 2\hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill \\ \hfill & \hfill & \hfill 5& =\hfill & 5\hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \end{array}$

$\{\begin{array}{ccc}\begin{array}{l}-x+y=-1\\ -x+y=2\end{array}& & \begin{array}{l}\left(1\right)\\ \left(2\right)\end{array}\end{array}$
Write each equation in slope-intercept form.

$\begin{array}{rrrrrrrrrrr}\hfill \left(1\right)& \hfill & \hfill -x+y& \hfill =& -1\hfill & \hfill & \hfill \left(2\right)& \hfill & \hfill -x+y& \hfill =& 2\hfill \\ \hfill & \hfill & \hfill y& \hfill =& x-1\hfill & \hfill & \hfill & \hfill & \hfill y& \hfill =& x+2\hfill \end{array}$
Graph each of these equations.

These lines are parallel. This system has no solution. We denote this fact by writing inconsistent .

We are sure that these lines are parallel because we notice that they have the same slope, $m=1$ for both lines. The lines are not coincident because the $y$ -intercepts are different.

$\{\begin{array}{ccc}\begin{array}{l}-2x+3y=-2\\ -6x+9y=-6\end{array}& & \begin{array}{l}\left(1\right)\\ \left(2\right)\end{array}\end{array}$
Write each equation in slope-intercept form.

$\begin{array}{rrrrrrrrrrr}\hfill \left(1\right)& \hfill & \hfill -2x+3y& \hfill =& -2\hfill & \hfill & \hfill \left(2\right)& \hfill & \hfill -6x+9y& \hfill =& -6\hfill \\ \hfill & \hfill & \hfill 3y& \hfill =& 2x-2\hfill & \hfill & \hfill & \hfill & \hfill 9y& \hfill =& 6x-6\hfill \\ \hfill & \hfill & \hfill y& \hfill =& \hfill \frac{2}{3}x-\frac{2}{3}& \hfill & \hfill & \hfill & \hfill y& \hfill =& \hfill \frac{2}{3}x-\frac{2}{3}\end{array}$

Both equations are the same. This system has infinitely many solutions. We write dependent .