# How to Solve a System of Linear Equations Using the Graphing Method

By Kim Marquardt
graph image by Roman Sigaev from Fotolia.com

There are three kinds of possible solutions to a system of linear equations: one solution, no solution or infinite solutions. If the system has one solution, the two lines intersect at one point. To have an infinite number of solutions to the system, the lines would be exactly the same. A system that has no solution means that the two lines are parallel to each other and will never intersect.

Solve both equations for y, if not already in the form y = mx + b, where m equals the slope and b is the y-intercept. For example, if the system of equations is y – 5x = -1 and -3x + 2y = 12, solve the first equation for y by adding 5x to both sides to give the equation y = 5x -1. Solve the second equation for y by adding 3x to both sides and dividing both sides by 2 to yield the equation y = 3/2x + 6.

Draw a coordinate plane (x and y axis) on the graph paper.

Label each axis with numbers.

Start graphing the first equation by placing a dot on the y-intercept. In the example system, for the first equation, y = 5x – 1, the y-intercept is -1. So a dot would be placed at the ordered pair (0,-1).

Graph the slope of the first equation. The slope of the example equation y = 5x -1 is 5/1. (A whole number is always over 1.) To graph this slope, start at the y-intercept point and go up 5 and to the right 1. Put a dot at this point (1, 4).

Draw a straight line through the points using a ruler.

Start graphing the second equation on the same graph as the first one by placing a dot on the y-intercept. In the example system, for the second equation, y = 3/2x + 6, the y-intercept is 6. So a dot would be placed at the ordered pair (0, 6).

Graph the slope of the second equation. The slope of the example equation y = 3/2x + 6 is 3/2. To graph this slope start at the y-intercept point and go up 3 and to the right 2. Put a dot at this point (2, 9).

Draw a straight line through the points using a ruler.

Find the solution to the system of equations. For the example system of equations, the solution is (2, 9) because the two lines intersect at this point.