A set of equations with multiple variables which can be solved using a specific set of values.
Systems of Equations in the Real World
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, and there can be many such answers for any given system. Answers are generally written in the form of an ordered pair: $\left( x,y \right)$. Approaches to solving a system of equations include substitution and elimination as well as graphical techniques.
There are several practical applications of systems of equations. These are shown in detail below.
Planning an Event
A system of equations can be used to solve a planning problem where there are multiple constraints to be taken into account:
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. So, how many students and how many teachers are invited to the party?
First, we need to identify and name our variables. In this case, our variables are teachers and students. The number of teachers will be $T$, and the 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 have a system of equations that can be solved by substitution, elimination, or graphically. The solution to the system is $S=49$ and $T=7$.
This next example illustrates how systems of equations are used to find quantities.
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! Looking at her newly increased pile, Davis remarks, "Wow, you've got $29$ more potatoes than I do!" 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:
$K-5 = 3D$
$D+29 = K$
From here, substitution, elimination, or graphing will reveal that $K=41$ and $D=12$.
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. Or, you can substitute your answers into every equation and check that they result in accurate solutions.
There are a multitude of other applications for systems of equations, such as figuring out which landscaper provides the best deal, how much different cell phone providers charge per minute, or comparing nutritional information in recipes.