Objectives
-
Solve questions involving applications of systems of equations.
Section 3.7 Applications of Systems of Linear Equations (LF7)
Subsection 3.7.1 Activities
Remark 3.7.1.
Now that we have explored multiple methods for solving systems of linear equations, let’s put those in to practice using some real-world application problems.
Activity 3.7.2.
Let’s begin by revisiting the carnival admission problem from Section 3.6.
Admission into a carnival for \(4\) children and \(2\) adults is \(\$128.50\text{.}\) For \(6\) children and \(4\) adults, the admission is \(\$208\text{.}\) Assuming a different price for children and adults, what is the price of the child’s admission and the price of the adult admission?
(a)
First, set up a system of equations representing the given information. Use \(x\) to represent the child admission price and \(y\) for the adult admission price.
-
\(\displaystyle \begin{cases} x+y=128.50\\ x+y=208 \end{cases}\)
-
\(\displaystyle \begin{cases} 2x+4y=128.50 \\ 4x+6y=208 \end{cases}\)
-
\(\displaystyle \begin{cases} 4x+2y=128.50\\ 6x+4y=208 \end{cases}\)
-
\(\displaystyle \begin{cases} 6x+4y=128.50 \\ 4x+2y=208 \end{cases}\)
(b)
Solve the system of equations.
-
\(\displaystyle (27, 10.25)\)
-
\(\displaystyle (15.25,24.5)\)
-
\(\displaystyle (24.5, 15.25)\)
-
\(\displaystyle (10,37)\)
(c)
Write your solution in terms of the price of admission for children and adults.
Activity 3.7.3.
Ammie’s favorite snack to share with friends is candy salad, which is a mixture of different types of candy. Today she chooses to mix Nerds Gummy Clusters, which cost \(\$8.38\) per pound, and Starburst Jelly Beans, which cost \(\$7.16\) per pound. If she makes seven pounds of candy salad and spends a total of \(\$55.61\text{,}\) how many pounds of each candy did she buy?
(a)
Set up a system of equations to represent the mixture problem. Let \(N\) represent the pounds of Nerds Gummy Clusters and \(S\) represent the pounds of Starburst Jelly Beans in the mixture.
-
\(\displaystyle \begin{cases} N+S=7 \\ 7.16N+8.38S=55.61 \end{cases}\)
-
\(\displaystyle \begin{cases} N+S=7 \\ 8.38N+7.16S=55.61 \end{cases}\)
-
\(\displaystyle \begin{cases} N+S=55.61 \\ 8.38N+7.16S=389.27 \end{cases}\)
-
\(\displaystyle \begin{cases} N+S=7 \\ 7N+7S=55.61 \end{cases}\)
(b)
Now solve the system of equations and put your answer in the context of the problem.
-
Ammie bought \(2.5\) lbs of Nerds Gummy Clusters and \(4.5\) lbs of Starburst Jelly Beans.
-
Ammie bought \(3.5\) lbs of Nerds Gummy Clusters and \(3.5\) lbs of Starburst Jelly Beans.
-
Ammie bought \(4.5\) lbs of Nerds Gummy Clusters and \(2.5\) lbs of Starburst Jelly Beans.
-
Ammie bought \(5.5\) lbs of Nerds Gummy Clusters and \(1.5\) lbs of Starburst Jelly Beans.
Activity 3.7.4.
A couple has a total household income of \(\$104{,}000\text{.}\) The wife earns \(\$16{,}000\) less than twice what the husband earns. How much does the wife earn?
(a)
Set up a system of equations to represent the situation. Let \(w\) represent the wife’s income and \(h\) represent the husband’s income.
-
\(\displaystyle \begin{cases} w+h=104000 \\ h=2w-16000 \end{cases}\)
-
\(\displaystyle \begin{cases} w+h=104000 \\ 2h=w-16000 \end{cases}\)
-
\(\displaystyle \begin{cases} w+2h=104000 \\ w=h-16000 \end{cases}\)
-
\(\displaystyle \begin{cases} w+h=104000 \\ w=2h-16000 \end{cases}\)
(b)
Solve the system of equations. How much does the wife earn?
-
The wife earns \(\$29{,}300\text{.}\)
-
The wife earns \(\$40{,}000\text{.}\)
-
The wife earns \(\$64{,}000\text{.}\)
-
The wife earns \(\$74{,}600\text{.}\)
Activity 3.7.5.
Kenneth currently sells suits for Company A at a salary of \(\$22{,}000\) plus a \(\$10\) commission for each suit sold. Company B offers him a position with a salary of \(\$28{,}000\) plus a \(\$4\) commission for each suit sold. How many suits would Kenneth need to sell for the options to be equal?
Set-up and solve a system of equations to represent the situation.
Subsection 3.7.2 Exercises
Exercises available at
https://library.tbil.org/2025/precalculus/exercises/#/bank/LF7/
