Question

A graph is shown. The x-axis is labeled "Months" and counts by 1s. The y-axis is labeled "Amount in Dollars" and counts by 175s. The line passes through (15, 0) and (0, 2625)
Barton took out a car loan of $2,625. He makes monthly payments of $175. The graph represents the remaining balance after each payment. Which example models the correct slope?
A. m=\frac{2450-2625}{0-1}
B. m=\frac{525-700}{11-12}
C. m=\frac{1050-1225}{9-8}
D. m=\frac{9-8}{1050-1225}

Answer 1

To determine the slope represented by the graph of the remaining balance after Barton's monthly payments, we need to establish what the slope represents.

In this context:
- The slope \( m \) represents the change in the remaining loan balance (in dollars) over the change in time (in months). Since Barton is making monthly payments of $175, we would expect the slope to represent a decrease in the remaining balance.

The slope can be calculated using the formula:

\[
m = \frac{\text{change in } y}{\text{change in } x}
\]

Given that for each month (1 month later), the balance decreases by $175:

- The line moves down by 175 dollars for each increase of 1 month.

Now let's analyze the options provided:

A. \( m = \frac{2450 - 2625}{0 - 1} \)

This represents the change in balance from \( 2625 \) at month \( 0 \) to \( 2450 \) at month \( 1 \):

\[
m = \frac{-175}{-1} = 175
\]

B. \( m = \frac{525 - 700}{11 - 12} \)

This represents a change in balance from \( 700 \) at month \( 12 \) to \( 525 \) at month \( 11 \):

\[
m = \frac{-175}{-1} = 175
\]

C. \( m = \frac{1050 - 1225}{9 - 8} \)

This represents a change in balance from \( 1225 \) at month \( 8 \) to \( 1050 \) at month \( 9 \):

\[
m = \frac{-175}{1} = -175
\]

D. \( m = \frac{9 - 8}{1050 - 1225} \)

This represents a reverse calculation that does not accurately reflect the balance change:

\[
m = \frac{1}{-175} = -\frac{1}{175}
\]

Both options A and B yield a correct slope of 175, which represents the correct change in balance for each month.

Thus, **both A and B model the correct slope.**

If you need to choose a single option, then either one would be acceptable as they are both accurate in representing the decrease in loan balance per month. To select only one, **A** is sufficient as it clearly illustrates the calculation for the first month. However, **B** is also valid for an understanding of how the next month relates.