Question

1. steph purchases a new laptop. to pay for the laptop, he will make a one - time down payment and then monthly payments of $25 each month. the linear relationship between the total amount he has paid, y, and the number of months since his purchase, x, is shown on the graph.

a. the point (9, 365) is identified on the graph. explain the meaning of this point in the context of the problem.
b. write an equation for the linear relationship.
c. if the laptop costs $540, how many months will it take for steph to pay off the laptop?

Explanation:

Part a

In the context of the problem, the point \((x, y)=(9, 365)\) means that \(x = 9\) represents the number of months since the purchase, and \(y=365\) represents the total amount paid. So after 9 months, Steph has paid a total of \$365 for the laptop.

Step 1: Recall the slope - intercept form of a linear equation

The slope - intercept form of a linear equation is \(y=mx + b\), where \(m\) is the slope (rate of change) and \(b\) is the y - intercept (initial value, in this case, the down - payment). We know that the monthly payment is \$25, so the slope \(m = 25\) (since for each increase of 1 in \(x\) (number of months), \(y\) (total amount paid) increases by 25). So the equation is \(y = 25x + b\).

Step 2: Find the y - intercept \(b\)

We know that the point \((9,365)\) lies on the line. Substitute \(x = 9\) and \(y=365\) into the equation \(y=25x + b\):
\[

$$\begin{align*} 365&=25\times9 + b\\ 365&=225 + b\\ b&=365 - 225\\ b&=140 \end{align*}$$

\]

Step 3: Write the final equation

Substitute \(m = 25\) and \(b = 140\) into the slope - intercept form \(y=mx + b\). We get \(y=25x + 140\).

Step 1: Set up the equation

We know that the total cost of the laptop \(y = 540\) and the equation of the line is \(y=25x + 140\). Substitute \(y = 540\) into the equation:
\[
540=25x + 140
\]

Step 2: Solve for \(x\)

Subtract 140 from both sides of the equation:
\[

$$\begin{align*} 540-140&=25x+140 - 140\\ 400&=25x \end{align*}$$

\]
Then divide both sides by 25:
\[
x=\frac{400}{25}=16
\]

Answer:

After 9 months, Steph has paid a total of \$365 for the laptop.

Part b