# Applications of Systems of Equations

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.

Systems of equations are extremely useful in applications where there is more than one unknown. We will explore applications that involve systems of equations and look at how to set up a system of equations with given information.

## System of Equations

A system of equations is a group of two or more equations containing the same variables. In a system of equations there is more than one unknown since the equations contain more than one variable. We can use these systems to solve for all of the variables, or unknowns, in the system.

Since a system of equations is a set of equations, we can also represent the system graphically by graphing all of the equations in the system on the same graph. For example, consider the following system.

3x - y = 1

4x + 2y = 8

The image shows this system represented graphically.

Notice that the intersection point of the equations on the graph is a point that satisfies both of the equations in the system. We call that point a solution to the system. The solution to a system of equations consists of values of the variables that make all of the equations in the system true. We see that a solution to the system in our example is x = 1 and y = 2, which can also be written as the ordered pair (1, 2).

There are multiple ways of finding solutions to systems of equations, but those strategies are for another lesson. In this lesson, we want to look at applications of systems of equations and how to set up a system of equations that we can use to solve a problem.

## Applications of Systems of Equations

Suppose you want to know the win/loss record of your school's basketball team. You know they played 24 games during the season, and you also know that they won 4 more games than they lost. We are looking for the number of wins and the number of losses, so we have two unknowns. Hmmm, more than one unknown, that should ring a bell!

Systems of equations are used to solve applications when there is more than one unknown and there is enough information to set up equations in those unknowns. In general, if there are n unknowns, we need enough information to set up n equations in those unknowns. When we have those two things, setting up a system of equations is a good way to begin to tackle the problem.

Finding your team's win/loss record involves finding more than one unknown, and we are given information to set up equations in those unknowns, so let's go ahead and set up a system of equations to represent this problem.

The first thing we want to do is represent our unknowns using variables. Let's let w = number of wins and l = number of losses. We are given that the team played 24 games total. We know that the number of wins plus the number of losses has to equal the total number of games played. Therefore, w + l = 24. We have our first equation.

Since there are two unknowns, we know we want one more equation. We are told that the team won 4 more games than they lost. This tells us that the number of losses plus 4 would give the number of wins. Putting that in equation form, we have that l + 4 = w. We have our second equation, so we have our system of equations.

w + l = 24

l + 4 = w

The image shows this system represented graphically.

We see that the intersection point is (14, 10). Thus, the solution to the system of equations is w = 14 and l = 10. We have the answer to our problem. Your school's basketball team won 14 games and lost 10 games. Not a bad season!

## Example

Let's consider one more example. Suppose you want to find three numbers given the following information.

• If you add all the numbers together you get 50.
• Two times the first number plus the second number is equal to the third number plus 22.
• Doubling the sum of the first and second number gives 3 times the third number.

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