# Solving Systems of Linear Equations Using Graphing

A system of linear equations is just a set of two or more linear equations.

In two variables $\left(x\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y\right)$ , the graph of a system of two equations is a pair of lines in the plane.

There are three possibilities:

• The lines intersect at zero points. (The lines are parallel.)
• The lines intersect at exactly one point. (Most cases.)
• The lines intersect at infinitely many points. (The two equations represent the same line.)

How to Solve a System of Equations Using the Graphing Method

This method is useful when you just need a rough answer, or you're pretty sure the intersection happens at integer coordinates. Just graph the two lines, and see where they intersect!

Example:

Solve the system by graphing.

$\begin{array}{l}y=0.5x+2\\ y=-2x-3\end{array}$

The two equations are in slope-intercept form.

The first line has a slope of $0.5$ and a $y$ -intercept of $2$ .

The second line has a slope of $-2$ and a $y$ -intercept of $-3$ .

Graph the two lines as shown.

The solution is where the two lines intersect, the point $\left(-2,1\right)$ . That is, $x=-2$ and $y=1$ .