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.