Solving a System of Linear Equations by Graphing

If you have a system of two linear equations, you can solve it by graphing. Here are the steps.

1.  Solve both equations for y to put them into slope-intercept form.
2.  Graph both equations on the same plane.
3.  Find the point where the lines intersect. This is the solution to the system of equations.

For a system of two linear equations, the lines could intersect once, at one point, for one solution; infinitely many times, if they are the same line, for infinitely many solutions (all real numbers); or not at all, if they are parallel lines, for no solution. Be aware of those possibilities.

Here is a video explaining how to solve a system of linear equations by graphing.



Graphing a line

In case you forgot how to graph a line since 055, here is a quick refresher…

Step 1: Solve the equation for y. You want your equation to look like


where m is the slope of the line and b is the y-intercept.

Step 2: Plot the y-intercept. The y-intercept is the point (0, b).  Note that b is a signed number: if is negative, then your y-intercept will be below the x-axis.

Step 3: Use the slope m to plot the next two points. Your slope might be a fraction. If it is not a fraction, it is best to turn it into one by putting the whole number slope over 1. The numerator (top number) of the fraction is the rise. The denominator (bottom number) is the run.

Start at the y-intercept (0, b) and move up the number of units of the rise and, from there, right the number of units of the run. Plot the point where you end up. Repeat the process to plot a third point.

Note: If m is negative, then make the rise negative. If the rise is negative, then you will go down that many units instead of up. You will still move to the right by the number of the run.

Step 4: When you have plotted three points, connect them. They should form a straight line. If they do not, you made a mistake.

Below the break is a video tutorial with two examples.