Steph purchases a new laptop. To pay for the laptop, he will make a one-time down payment and then monthly payments of \$25 $ 25 $25each month. The linear relationship between the total amount he has paid, y 𝑦 y, and the number of months since his purchase, x 𝑥 x, is shown on the graph. ﻿ a. The point (9,365) ( 9 , 365 ) (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 $ 540 $540, how many months will it take for Steph to pay off the laptop?

Let's go through each part of the problem step-by-step.

### Part (a): Explain the Meaning of the Point (9, 365)

The point $ (9, 365) $ on the graph represents a key relationship between the number of months and the total amount paid towards the laptop. Here's what the coordinates mean:

- $9$ represents the number of months after the purchase.
- $365$ represents the total amount Steph has paid after 9 months.

Thus, the point $ (9, 365) $ means that **9 months after purchasing the laptop, Steph has paid a total of \$365.**

### Part (b): Write an Equation for the Linear Relationship

The linear relationship between the total amount paid ($y$) and the number of months since purchase ($x$) follows the standard equation of a line:

\[
y = mx + b
\]

Where:
- $y$ is the total amount paid after $x$ months,
- $m$ is the slope (which represents the monthly payment),
- $b$ is the $y$-intercept (which represents the initial payment or down payment).

#### Step 1: Find the Slope ($m$)

The slope $m$ represents the change in the total amount paid per month. In this case, Steph makes a monthly payment of \$25, so the slope is:

\[
m = 25
\]

#### Step 2: Find the $y$-intercept ($b$)

To find the $y$-intercept, we can use the point $ (9, 365) $ (since it lies on the line) and substitute the values of $x = 9$ and $y = 365$ into the equation:

\[
y = mx + b
\]

Substitute $m = 25$, $x = 9$, and $y = 365$:

\[
365 = 25(9) + b
\]

Simplify:

\[
365 = 225 + b
\]

Solve for $b$:

\[
b = 365 - 225 = 140
\]

So, the $y$-intercept is $b = 140$, meaning Steph made an initial down payment of **\$140**.

#### Step 3: Write the Equation

Now that we have both the slope ($m = 25$) and the $y$-intercept ($b = 140$), the equation of the linear relationship is:

\[
y = 25x + 140
\]

This equation represents the total amount paid ($y$) after $x$ months.

### Part (c): How Many Months Will It Take for Steph to Pay Off the Laptop?

The laptop costs \$540, and we need to determine how many months it will take for Steph to pay off the laptop, meaning the total amount paid $y = 540$.

Substitute $y = 540$ into the equation:

\[
540 = 25x + 140
\]

Solve for $x$:

\[
540 - 140 = 25x
\]

\[
400 = 25x
\]

\[
x = \frac{400}{25} = 16
\]

Thus, it will take Steph **16 months** to pay off the laptop.

### Summary

- **Part (a):** The point $ (9, 365) $ means that 9 months after purchasing the laptop, Steph has paid a total of \$365.
- **Part (b):** The equation for the linear relationship is $ y = 25x + 140 $.
- **Part (c):** It will take Steph 16 months to pay off the laptop.

Would you like more details or have any questions? Here are some related questions for further exploration:

1. How much has Steph paid after 12 months?
2. What is the total amount paid after 24 months?
3. If the laptop cost \$600, how many months would it take to pay it off?
4. How would the equation change if Steph made monthly payments of \$30 instead of \$25?
5. How can we represent this payment plan graphically, and what would the slope and intercepts signify?

**Tip:** Linear equations like this one are perfect for modeling consistent payments or growth over time. Understanding how to interpret slope and intercept helps in solving real-life financial problems.

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?

Learn how to solve a real-world problem involving a linear equation that models monthly payments for a laptop purchase. Understand the meaning of graph points, create a linear equation, and calculate the time required to pay off the laptop. Suitable for students in Grades 6-8.

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