Chapter 8, Problem 14CST

a.

To determine

To write:

An equation describing the possible number of movies and video games you can rent.

Answer to Problem 14CST

The equation $4x+5y=100$ represents the possible number of movies and video games you can rent.

Explanation of Solution

Given:

You have a $100 gift card to a store that rents movies and video games. The rental fee for a movie is $4 and the rental fee for a video game is $5.

Calculation:

Let x represent number of movies rented and y represent number of video games rented.

We have been given that the rental fee for a movie is $4, so rental fee for x movies would be $4x$ .

We are also told that the rental fee for a video game is $5, so rental fee for y video games would be $5y$ .

The total cost of renting x movies and y video games would be equal to $100. We can represent this information in an equation as:

  $4x+5y=100$

Therefore, the equation $4x+5y=100$ represents the possible number of movies and video games you can rent.

b.

To determine

To graph:

Equation from part (a) using the intercepts.

Answer to Problem 14CST

The graph of equation $4x+5y=100$ would look like:

  

Explanation of Solution

Given:

You have a $100 gift card to a store that rents movies and video games. The rental fee for a movie is $4 and the rental fee for a video game is $5.

Calculation:

To find x-intercept, we will substitute $y=0$ in our given equation and solve for xas shown below:

  $\begin{array}{l}4x+5y=100\left(\text{Given equation}\right)\\ 4x+5(0)=100\left(\text{Substituting }y=0\right)\\ 4x=100\\ \frac{4x}{4}=\frac{100}{4}\left(\text{Dividing both sides by 4}\right)\\ x=25\end{array}$

Therefore, the x-intercept of our given equation is $(25,0)$ .

To find y-intercept, we will substitute $x=0$ in our given equation and solve for yas shown below:

  $\begin{array}{l}4x+5y=100\left(\text{Given equation}\right)\\ 4(0)+5y=100\left(\text{Substituting }x=0\right)\\ 0+5y=100\\ 5y=100\\ \frac{5y}{5}=\frac{100}{5}\\ y=20\end{array}$

Therefore, the y-intercept of our given equation is $(0,20)$ .

Upon graphing our given equation, we will get our required graph as shown below:

  

c.

To determine

To write:

Three possible combinations of movies and video games you can rent.

Answer to Problem 14CST

Three possible combinations would be:

1. (5,16)

2. (10,12)

3. (15,8)

Explanation of Solution

Given:

You have a $100 gift card to a store that rents movies and video games. The rental fee for a movie is $4 and the rental fee for a video game is $5.

Calculation:

To find some possible combinations, we will substitute some values of x in our given equation as shown below:

  $\begin{array}{l}4x+5y=100\left(\text{Given equation}\right)\\ 4(5)+5y=100\left(\text{Substituting }x=5\right)\\ 20+5y=100\\ 5y=80\left(\text{Subtracting 20 from both sides}\right)\\ y=16\left(\text{Dividing both sides by 5}\right)\end{array}$

  $\begin{array}{l}4x+5y=100\left(\text{Given equation}\right)\\ 4(10)+5y=100\left(\text{Substituting }x=10\right)\\ 40+5y=100\\ 5y=60\left(\text{Subtracting 60 from both sides}\right)\\ y=12\left(\text{Dividing both sides by 5}\right)\end{array}$

  $\begin{array}{l}4x+5y=100\left(\text{Given equation}\right)\\ 4(15)+5y=100\left(\text{Substituting }x=15\right)\\ 60+5y=100\\ 5y=40\left(\text{Subtracting 60 from both sides}\right)\\ y=8\left(\text{Dividing both sides by 5}\right)\end{array}$

Therefore, the three possible combinations would be (5,16), (10,12) and (15,8).