# Solving a System of Absolute Value Equations by Graphing

Instructor: David Karsner
A system of absolute value equations is two more equations that contain absolute value symbols. To solve these systems you will need to find a point (x,y) or points that will work for both equations. This lesson explains how to solve these systems by graphing.

## From Linear to Absolute Value

If you are familiar with the graphs of functions, you will recall that the graph of a linear function is a line. The point at which the line crosses the x-axis is called the x-intercept. Imagine breaking that line at the x-intercept and using the broken off portion to create a V with the portion above the x-intercept. You have just created the graph of an absolute value equation. If you have more than one equation on the same coordinate grid, you can find the points that are included in both of the graphs. This lesson is about how to solve absolute value systems by graphing these V's and finding where the graphs intersect.

## Absolute Value Equations

Absolute value is how far away something is from zero. For example, -4 and 4 are both 4 units away from zero. Absolute value is represented by two parallel line segments one on either side of the expression. Equations can contain absolute value symbols. A simple example would be the (AbsVal x) = 5. What value of x will give you an absolute value of 5? The answers are 5 and -5.

## A System of Equations

A system of equations is two or more equations that are to be solved at the same time. The solution to a system is the point or points that make all of the equations true. The two equations x + y = 7 and x - y = 1 would be a system of equations. The point (4,3) would be a solution. 4 + 3 = 7 and 4 - 3 = 1. The point (4,3) works for both equations. A system of absolute value equations is two or more equations that contain absolute value symbols and are to be solved together.

## Using the Graphs to Solve

There are several ways to solve a system of absolute value equations. In this lesson, we will solve the system by graphing the two absolute value equations on the same coordinate grid. The point or points where the graphs of the equations intersect will be the solution to the system of equations. For this lesson, we will be looking at absolute value equations that are linear, and the entire equation is inside of the absolute value bars. The graph of the basic equation absolute value of x will be a V with the vertex on (0,0) and having a slope of one to the right and negative one to the left.

Since all of our equations will be inside of the absolute value bars, all the graphs will have their vertex on the x-axis. Both equations of the system will be of the form ax + b where the positive version of a is the slope of the right portion of the V and the negative version of a is the slope of the left side portion of the V. The vertex of the graph is the point of the V. It is located at the x value that causes ax + b = 0.

• For example, the absolute value (3/2)x - 6
• has a vertex at (3/2) x - 6 = 0
• (3/2) x = 6
• 3x = 12
• x = 4.

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