Solve a system of equations by graphing

A system of equations is a set of two or more equations. The solution is the one point (x, y) that makes ALL the equations true. Graphing is a visual way to find this solution—it's the point where the lines intersect!

Do this: Read the concept below, then try the quiz or activity.

Lesson 133 of 188

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Concept

To solve a system of linear equations by graphing, you graph each line on the same coordinate plane and find the point where they cross.

The Solution: The solution to a system of linear equations is the point of intersection. This (x, y) coordinate is the only point that lies on both lines, meaning it is the only point that satisfies both equations.

Steps to Solve by Graphing:

1.  Write both equations in slope-intercept form (y = mx + b). This makes them easy to graph.
2.  Graph the first equation.
    *   Start by plotting the y-intercept (the 'b' value) on the y-axis.
    *   From the y-intercept, use the slope (m = rise/run) to find a second point.
    *   Draw a straight line through these two points.
3.  Graph the second equation on the same coordinate plane using the same method.
4.  Find the point of intersection. This is the (x, y) coordinate where the two lines cross.
5.  Check your solution. Plug the x and y values from the intersection point into both of the original equations to make sure they are both true.

Example: Solve the system y = 2x - 1 and y = -x + 5

1.  Slope-intercept form: Both equations are already in this form.
2.  Graph y = 2x - 1:
    *   Y-intercept (b) is -1. Plot a point at (0, -1).
    *   Slope (m) is 2, or 2/1. From (0, -1), rise 2 and run 1 to find a second point at (1, 1).
    *   Draw the line.
3.  Graph y = -x + 5:
    *   Y-intercept (b) is 5. Plot a point at (0, 5).
    *   Slope (m) is -1, or -1/1. From (0, 5), go down 1 and right 1 to find a second point at (1, 4).
    *   Draw the line.
4.  Find intersection: The lines cross at the point (2, 3).
5.  Check:
    *   In y = 2x - 1: Does 3 = 2(2) - 1? 3 = 4 - 1. Yes.
    *   In y = -x + 5: Does 3 = -(2) + 5? 3 = 3. Yes.
*   The solution is correct.

Types of Solutions:
*   One Solution: The lines intersect at one point (most common).
*   No Solution: The lines are parallel and never intersect. They have the same slope but different y-intercepts.
*   Infinitely Many Solutions: The two equations represent the exact same line. They have the same slope and the same y-intercept.

Try it

Practice: Solve a system of equations by graphing.

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