# Applications of 2x2 Systems of Equations

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will quickly review what a 2x2 system of equations is, and then we will look at how to apply these systems to real world problems. We will see how to set up a system to correspond to a problem and discuss solutions.

## 2x2 System of Equations

Before getting to applications of 2x2 systems of equations, let's have a quick review of vocabulary. A system of equations is a set of two or more equations with the same variables. For example, the following is a system of equations.

2x - 5y = 15

3x + y = 31

In this particular example, there are two equations and two variables, therefore, we call this a 2x2 system of equations.

In general, an n x n system of equations has n equations and n variables.

A solution to a system of equations consists of values of the variables that make all of the equations in the system true, and the set of all solutions to a system is called the solution set of the system. Consider our example above. If we plug in x = 10 and y = 1 into each of the equations, both of the equations are true.

2(10) - 5(1) = 15

3(10) + 1 = 31

Therefore, x = 10 and y = 1 is a solution to the system. We can also represent this solution as the ordered pair (10,1).

## Applications Solved Using a 2x2 System of Equations

As we've seen, a 2x2 system of equations has two variables, so there are two unknowns. Since there are two unknowns, these systems come in handy when we have a problem where we need to find two different values. When we have two unknowns and enough information in a problem to set up two equations in those unknowns, we can use a 2x2 system of equations to solve the problem.

For instance, suppose you are working at a health food store, and you need to mix almonds worth \$5/lb. with cashews worth \$9/lb. to make 10 lbs. of a mixture worth \$7/lb. Notice that there are two unknowns in this situation. One is the number of pounds of almonds to put in the mixture, and the other is the number of pounds of cashews to put in the mixture. Let's represent these unknowns with variables.

a = number of pounds of almonds

c = number of pounds of cashews

We know that we have 10 lbs. in the mixture all together, so a + c = 10. We also know that each pound of almonds cost \$5, so the total cost of almonds in the mixture is 5a. Similarly, the total cost of cashews in the mixture is 9c. Since there are 10lbs. of mixture, and it costs \$7 per pound, the entire mixture costs \$70. We can put these three facts together to get the equation 5a + 9c = 70. We now have two equations and two unknowns. In other words, we have a 2x2 system of equations.

a + c = 10

5a + 9c = 70

There are many ways to go about solving a 2x2 system, and each of these would take a whole lesson to explain. Thus, in this lesson, we will just concentrate on the applications and their solutions. In this nut example, the solution to the system we set up is a = 5 and c = 5. If we plug these values into our system, both of the equations are true.

5 + 5 = 10

5(5) + 9(5) = 70

Therefore, to make the desired mixture, we need to add 5 lbs. of the almonds and 5 lbs. of the cashews.

## Another Application Example

Have you ever seen one of those contests asking how many jellybeans are in a jar? Imagine you come across a contest that has you guessing the number of red jellybeans and the number of green jellybeans in a jar. You are given the following facts.

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