How to Solve Linear Equations By Graphing

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

This lesson will give a quick review of systems of linear equations and their solutions. We will then look at taking an example through a step-by-step process that allows us to solve a system of linear equations by graphing.

System of Linear Equations

Suppose you are selling tickets to a fundraiser for a cause that is very important to you. A ticket costs $15, but there is a $3 military discount for soldiers, so their tickets cost $12. After you are done selling tickets, you've sold 190 tickets, and you've collected $2,520. You need to know how many tickets you've sold to soldiers and how many tickets you've sold to non-soldiers. Any ideas on how to do this?

One way of solving this problem lies in a system of linear equations. A system of linear equations is a set of two or more linear equations with the same variables, where a linear equation is an equation that is a line when graphed. Systems of linear equations are used to solve problems when there is more than one unknown. A solution to a set of linear equations consists of values of the variables in the system that makes all of the equations in the system true. A solution set is the set of all of the solutions to a system of equations. Let's consider our fundraiser example to better understand this.

If we let x be the number of non-soldier tickets sold and y be the number of soldier tickets sold, we can set up two equations in these variables and have a system of equations representing our problem. We know that you sold 190 tickets all together. This gives that x + y = 190. We also know that you collected $2,520 while charging $15 for each non-soldier ticket and $12 for each soldier ticket. This gives that 15x + 12y = 2520. We have the following system of equations:


sysgraph1


Now, we just want to find a solution to this system so we know how many of each type of ticket we've sold.

Solving a System of Linear Equations by Graphing

There are many different ways to go about solving a system of linear equations. In this lesson, we are going to look at solving by graphing. As we said, a solution to a system of linear equations consists of values of the variables that make all of the equations in the system true. Once again, consider our fundraiser example.

To solve this system by graphing, we use the fact that if we have values of x and y that make each of the equations true, then the point (x, y) is on the graphs of each of the equations in the system. Also, if a point is on the graph of both of the equations in this system, then it must be at a point where their graphs intersect.

Keeping this in mind, we have the following steps to solve a system of linear equations by graphing:

  1. Graph each of the equations in the system on the same coordinate axes.
  2. Find the points at which the graphs intersect. These points are the solutions to the system.
  3. Plug the points back into each of the equations to verify that it makes all of the equations in the system true.

Let's take our fundraiser example through these steps to solve the system.

Solving the Fundraiser Example by Graphing

The first thing we want to do is graph each of the equations in the system on the same coordinate axes.


Step 1
sysgraph2


The second step is to identify the intersection points of the graphs.


Identify Intersection Point
sysgraph3


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