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Use the image to answer the question.

A line with an arrow at the bottom is plotted in quadrant 1 of a coordinate plane. The x-axis is labeled Months, and the y-axis is labeled Amount in Dollars.

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?

(1 point)
Responses

m=525−70011−12
m equals Start Fraction 525 minus 700 over 11 minus 12 End Fraction

m=1,050−1,2259−8
m equals Start Fraction 1,050 minus 1,225 over 9 minus 8 End Fraction

m=2,450−2,6250−1
m equals Start Fraction 2,450 minus 2,625 over 0 minus 1 End Fraction

m=9−81,050−1,225

1 answer

To find the correct slope for the scenario where Barton has a car loan of $2,625 and makes monthly payments of $175, we need to look at how the remaining balance changes with each payment.

Understanding the situation:

  • Initial balance (at month 0): $2,625
  • Monthly payment: $175
  • Remaining balance decreases by $175 each month.

Slope calculation:

The slope \( m \) can be calculated using the formula:

\[ m = \frac{\text{change in y}}{\text{change in x}} = \frac{\text{remaining balance after payment}}{\text{number of months}} \]

We can set up the slope between two points. Let’s suppose we take points for month 1 and month 2:

  1. After month 1:
    • Balance: \( 2625 - 175 = 2450 \)
  2. After month 2:
    • Balance: \( 2450 - 175 = 2275 \)

Points to consider for the slope:

  • Point at month 1: \( (1, 2450) \)
  • Point at month 2: \( (2, 2275) \)

Using these points to calculate the slope:

\[ m = \frac{2275 - 2450}{2 - 1} = \frac{-175}{1} = -175 \]

Evaluating the options:

Now we will evaluate the provided options to find which one correctly represents the slope of the line in this context:

  1. Option 1: \[ m = \frac{525 - 700}{11 - 12} \quad \text{(wrong)} \]

  2. Option 2: \[ m = \frac{1050 - 1225}{9 - 8} \quad \text{(wrong)} \]

  3. Option 3: \[ m = \frac{2450 - 2625}{0 - 1} = \frac{-175}{-1} = 175 \quad \text{(wrong)} \]

  4. Option 4: \[ m = \frac{9 - 8}{1050 - 1225} \quad \text{(confusing)} \]

None of the provided options directly stated a monthly slope of \(-175\) in a clear, logical way. However, the context most closely resembles Option 3, considering that the slope should remain consistent.

Therefore, none of the options perfectly represent the slope derived from the loan's monthly balance changes, but the rate of change is definitely \( -175 \), reflecting how each month the loan amount decreases.

It seems we might be looking to select one of the options fundamentally incorrect in presentation, much like misconstruing how the question was framed.

Given the structure, if forced to select, I would suggest Option 3 for having the intended numbers involved, but it remains fundamental to appreciate the exact relationships used to calculate negative impacts of loan payments over time.