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Applications of Systems of Equations
Systems of equations can be used to solve many reallife problems in which multiple constraints are used on the same variables.
Learning Objective

Apply systems of equations in two variables to real world examples
Key Points
Term

system of equations
A set of equations with multiple variables which can be solved using a specific set of values.
Full Text
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, and there can be many such answers for any given system. Answers are generally written in the form of an ordered pair:
There are several applications of systems of equations. This is shown in details below.
Planning an Event
The following example shows how a system of equations is used to solve a planning problem.
Emily is hosting a major afterschool 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 our variables. In this case, our variables are teachers and students. Now we need to name these variables: 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.
Finding Unknown Quantities
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 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
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.
Other Applications
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 provider charges for each minute, or comparing nutritional information in recipes.
Assign just this concept or entire chapters to your class for free.
Key Term Reference
 constraint
 Appears in these related concepts: Relative Minima and Maxima, Applications and Mathematical Models, and Application of Systems of Inequalities: Linear Programming
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Graphs of Equations as Graphs of Solutions
 function
 Appears in these related concepts: Solving Differential Equations, Visualizing Domain and Range, and The Vertical Line Test
 graph
 Appears in these related concepts: Reading Points on a Graph, Graphing Equations, and Graphical Representations of Functions
 principal
 Appears in these related concepts: MultiPeriod Investment, Types of Bonds, and Formulas and ProblemSolving
 set
 Appears in these related concepts: Sequences of Mathematical Statements, Sequences, and Introduction to Sequences
 solution
 Appears in these related concepts: Electrolyte and Nonelectrolyte Solutions, Turning Your Claim Into a Thesis Statement, and What is an Equation?
 unknown
 Appears in these related concepts: Linear Equations and Their Applications, Models Involving Nonlinear Systems of Equations, and Introduction to Variables
 variable
 Appears in these related concepts: Fundamentals of Statistics, What is a Linear Function?, and Math Review
Sources
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Source: Boundless. “Applications of Systems of Equations.” Boundless Algebra. Boundless, 25 Jul. 2016. Retrieved 27 Jul. 2016 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/systemsofequations3/systemsofequationsintwovariables40/applicationsofsystemsofequations1932244/