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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 $(x and y)$ , 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.

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

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.

Math diagram

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