Question

1. you are purchasing a new car. the car cost $19,200 and you will pay $320 a month.

a. write an equation that relates how much you owe on the car (y) to the number of months (x).
b. how many months will it take you to pay off the car?
c. what does the slope mean in the context of this problem? make sure you mention what each variable stands for and how the indicated value impacts the two variables.

Explanation:

Part (a)

Step1: Define variables and relationship

Let \( y \) be the amount owed, \( x \) be the number of months. The initial amount owed is the car cost (\( \$19200 \)), and each month we pay \( \$300 \), so the amount owed decreases by \( 300 \) per month. The equation is linear: \( y = -300x + 19200 \).

Step1: Set \( y = 0 \) (paid off)

To find when the car is paid off, set \( y = 0 \) in \( y = -300x + 19200 \). So \( 0 = -300x + 19200 \).

Step2: Solve for \( x \)

Add \( 300x \) to both sides: \( 300x = 19200 \). Divide both sides by \( 300 \): \( x=\frac{19200}{300}=64 \).

In the equation \( y = -300x + 19200 \), the slope is \( -300 \). Here, \( x \) is the number of months, \( y \) is the amount owed. The slope (\( -300 \)) means the amount owed (\( y \)) decreases by \( \$300 \) for each additional month (\( x \)) of payments. So each month, the debt reduces by \( \$300 \).

Answer:

\( y = -300x + 19200 \)

Part (b)