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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
Terms

constraint
A condition that a solution to a problem must satisfy.

system of equations
A set of equations with multiple variables which can be solved using a specific set of values.

ordered pair
A set containing exactly two elements in a fixed order, used to represent a point in a Cartesian coordinate system. Notation: (x, y).
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. 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 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. 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 .
Pendulum
This animation shows the velocity and acceleration vectors for a pendulum. One may note that at the maximum height of the pendulum's mass, the velocity is zero. This corresponds to zero kinetic energy and thus all of the energy of the pendulum is in the form of potential energy. When the pendulum's mass is at its lowest point, all of its energy is in the form of kinetic energy and we see its velocity vector has a maximum magnitude here.
Interactive Graph: Graphical Representation of a Systems of Equations
Graph of systems of equations with the equations and Once you have created your system of equations, one way to solve it is by showing them graphically, and then finding the intersecting point or points.
Key Term Reference
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 function
 Appears in these related concepts: Inverse Functions, Solving Differential Equations, and Functions and Their Notation
 graph
 Appears in these related concepts: Graphing on Computers and Calculators, Reading Points on a Graph, and Graphing Functions
 principal
 Appears in these related concepts: MultiPeriod Investment, Types of Bonds, and Maximizing Shareholder and Market Value
 set
 Appears in these related concepts: Sequences of Statements, Expressions and Sets of Numbers, and Executive Function and Control
 variable
 Appears in these related concepts: Related Rates, Math Review, and Psychology and the Scientific Method: From Theory to Conclusion
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Source: Boundless. “Applications of Systems of Equations.” Boundless Algebra. Boundless, 12 Aug. 2015. Retrieved 30 Aug. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/systemsofequationsandmatrices6/systemsofequationsintwovariables40/applicationsofsystemsofequations1932244/