Linear Equations: Solutions Using Graphing with Two Variables

Example 1

Solve this system of equations by graphing.

To solve using graphing, graph both equations on the same set of coordinate axes and see where the graphs cross. The ordered pair at the point of intersection becomes the solution (see Figure 1). 

Check the solution.

The solution is x = 3, y = –2. 

Figure 1. Two linear equations.

Solving systems of equations by graphing is limited to equations in which the solution lies close to the origin and consists of integers; even then, that solution is an approximation solved by eyeballing. For those reasons, graphing is used least frequently of all the solution methods.

Here are two things to keep in mind:

  • Dependent system. If the two graphs coincide—that is, if they are actually two versions of the same equation—then the system is called a dependent system, and its solution can be expressed as either of the two original equations. 
  • Inconsistent system. If the two graphs are parallel—that is, if there is no point of intersection—then the system is called an inconsistent system, and its solution is expressed as an empty set {}, or the null set, ⊘.