Chapter 7.2, Problem 75E

(a)

To determine

To create: a system of linear equations to find the number of adults $A$ and children $C$ at the game.

Answer to Problem 75E

The system of linear equations to find the number of adults $A$ and children $C$ at the game is given as:

  $\begin{array}{l}A+C=1175\\ 15A+12C=16275\end{array}$

Explanation of Solution

Given information:

Total number of people who attended baseball league $=1175$

Rate of ticket for adults $=\$15\text{ per person}$

Rate of ticket for children $=\$12\text{ per person}$

Total ticket revenue $=\$16275$

Calculation:

Create a system of linear equations to find the number of adults $A$ and children $C$ at the game.

According to the question

  $\begin{array}{l}\left[\begin{array}{l}\text{Total number of attendents=}\\ \text{Number of adults + number of childrens}\end{array}\right]\\ \text{1175=A+C}\end{array}$

And

  $\begin{array}{l}\text{Total ticket revenue}=16275\\ 15A+12C=16275\end{array}$

Therefore, the system of linear equations can be written as:

  $\begin{array}{l}A+C=1175\\ 15A+12C=16275\end{array}$

(b)

To determine

To solve: a system of linear equations to find the number of adults $A$ and children $C$ at the game by elimination method or substitution method and explain the choice of method.

Answer to Problem 75E

The number of adults $A$ and children $C$ at the game by elimination method are $725$ and $450$ respectively.

Explanation of Solution

Given information:

Total number of people who attended baseball league $=1175$

Rate of ticket for adults $=\$15\text{ per person}$

Rate of ticket for children $=\$12\text{ per person}$

Total ticket revenue $=\$16275$

Calculation:

Solve the system of linear equations by elimination method because in this method, either add or subtract the given equations to get an equation in one variable.

The system of linear equations can be written as:

  $A+C=1175$  ...(1)

  $15A+12C=16275$  ...(2)

Multiply equation (1) by $15$ :

  $15A+15C=17625$  ...(3)

Solve equation (2) and (3) by elimination method:

  $\underset{\_}{\begin{array}{l}\text{ 15A+}15C=17625\\ \text{- 15A}+12C=16275\end{array}}$

  $\begin{array}{l}\text{ 3C}=1350\\ \text{ C}=\frac{1350}{3}\\ \text{ C}=450\end{array}$

And

  $\begin{array}{l}A+C=1175\\ A=1175-C\\ A=1175-450\\ A=725\end{array}$

Therefore, the number of adults $A$ and children $C$ at the game by elimination method are $725$ and $450$ respectively.

(c)

To determine

To solve: a system of linear equations to find the number of adults $A$ and children $C$ at the game by graphing utility.

Answer to Problem 75E

The number of adults $A$ and children $C$ at the game by graphing utility are $725$ and $450$ respectively.

Explanation of Solution

Given:

Total number of people who attended baseball league $=1175$

Rate of ticket for adults $=\$15\text{ per person}$

Rate of ticket for children $=\$12\text{ per person}$

Total ticket revenue $=\$16275$

Calculation:

Create a system of linear equations to find the number of adults $A$ and children $C$ at the game.

According to the question

  $\begin{array}{l}\left[\begin{array}{l}\text{Total number of attendents=}\\ \text{Number of adults + number of childrens}\end{array}\right]\\ \text{1175=A+C}\end{array}$

And

  $\begin{array}{l}\text{Total ticket revenue}=16275\\ 15A+12C=16275\end{array}$

Therefore, the system of linear equations can be written as:

  $\begin{array}{l}A+C=1175\\ 15A+12C=16275\end{array}$

Use graphing utility to solve the given system of equations:

  

  $P\left(725,450\right)$ is the graphical solution of the given system of linear equations.

Therefore, the number of adults $A$ and children $C$ at the game by graphing utility are $725$ and $450$ respectively.