Question

4.5 worksheet

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: Identify the linear model

The relationship between the amount owed (\(y\)) and the number of months (\(x\)) is linear. The general form is \(y = mx + b\), where \(m\) is the slope (rate of change) and \(b\) is the initial value (y - intercept).

Step2: Determine the slope and y - intercept

The initial cost of the car (amount owed at \(x = 0\) months) is \(b=\$19200\). The monthly payment is \(\$320\), which means the amount owed decreases by \(\$320\) each month. So the slope \(m=- 320\) (negative because the amount owed is decreasing).

Step3: Write the equation

Substitute \(m = - 320\) and \(b = 19200\) into the linear equation \(y=mx + b\). We get \(y=-320x + 19200\).

Step1: Set up the equation for pay - off

To pay off the car, the amount owed \(y = 0\). So we set \(y = 0\) in the equation \(y=-320x + 19200\), which gives the equation \(0=-320x + 19200\).

Step2: Solve for \(x\)

First, subtract \(19200\) from both sides of the equation:
\(0-19200=-320x+19200 - 19200\)
\(-19200=-320x\)

Then, divide both sides by \(- 320\):
\(x=\frac{-19200}{-320}\)
\(x = 60\)

In the equation \(y=-320x + 19200\), \(x\) represents the number of months and \(y\) represents the amount owed on the car. The slope \(m=-320\). In the context of this problem, the slope means that for each increase of 1 in the number of months (\(x\)), the amount owed on the car (\(y\)) decreases by \(\$320\). This is because the slope of a linear equation \(y = mx + b\) represents the rate of change of \(y\) with respect to \(x\). Since the slope is - 320, it indicates a decrease of 320 dollars in the amount owed for each additional month of payments.

Answer:

\(y=-320x + 19200\)

Part (b)