## Applications of Systems of Equations

A system of equations, also known as simultaneous equations, is a set of equations that have multiple variables. The answer to a system of equations is a set of values that satisfies all equations in the system. Systems of equations can have multiple sets of answers that are correct. Solutions to a system of equations are often written as ordered pairs, (x,y). There are many ways of solving a system of equations, including the elimination, substitution, and graphical methods.

There are many applications of systems of equations. Whenever you have a problem that has multiple variables, setting up a system of equations is often the best method for solving. The steps you need to take in order to do that are: (1) identify the variables in the problem, (2) name the variables, and (3) set up the equations and solve for each variable.

### Example 1

Emily is hosting a major after-school party. The principal has imposed two restrictions. First, the total number of people attending (teachers and students combined) must be 56. Second, there must be one teacher for every seven students. How many students and how many teachers are invited to the party?

First, we need to identify our variables. In this case, our variables are teachers and students. Now we need to name these variables: number of teachers will be T, and number of students will be S.

Now we need to set up our equations. There is a constraint limiting the total number of people in attendance to 56, so:

For every seven students, there must be one teacher, so:

Now we a system of equations that can be solved by substitution, elimination, or graphically. The solution to the system is S=49 and T=7.

### Example 2

A group of 75 students and teachers are in a field, picking sweet potatoes for the needy. Kasey picks three times as many sweet potatoes as Davis—and then, on the way back to the car, she picks up five more sweet potatoes than that! Looking at her newly increased pile, Davis remarks "Wow, you've got 29 more potatoes than me! " How many sweet potatoes did Kasey and Davis each pick?

To solve, we first define our variables. The number of sweet potatoes that Kasey picks is K, and the number of sweet potatoes that Davis picks is D.

Now we can write equations based on the situation:

From here, substitution, elimination or graphing will reveal that K is 41 and D is 12.

It is important that you always check your answers. A good way to check solutions to a system of equations is to look at the functions graphically and then see where the graphs intersect .