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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.

y = 0.5 x + 2 y = − 2 x − 3

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 ( − 2 , 1 ) . That is, x = − 2 and y = 1 .