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
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:
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 afterschool party. The principal has imposed two restrictions. First, the total number of people attending (teachers and students combined) must be
First, we need to identify and name our variables. In this case, our variables are teachers and students. The number of teachers will be
Now we need to set up our equations. There is a constraint limiting the total number of people in attendance to
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
Finding Unknown Quantities
This next example illustrates how systems of equations are used to find quantities.
A group of
To solve, we first define our variables. The number of sweet potatoes that Kasey picks is
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. Or, you can substitute your answers into every equation and check that they result in accurate solutions.
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 providers charge per minute, or comparing nutritional information in recipes.
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: Visualizing Domain and Range, The Vertical Line Test, and Solving Differential Equations
 graph
 Appears in these related concepts: Graphical Representations of Functions, Graphing Equations, and Reading Points on a Graph
 principal
 Appears in these related concepts: MultiPeriod Investment, Types of Bonds, and Formulas and ProblemSolving
 set
 Appears in these related concepts: Sequences, Introduction to Sequences, and Sequences of Mathematical Statements
 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: Introduction to Variables, Models Involving Nonlinear Systems of Equations, and Linear Equations and Their Applications
 variable
 Appears in these related concepts: What is a Linear Function?, Fundamentals of Statistics, and Math Review
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