To determine the correct slope that models Barton's car loan payments, we need to understand that the slope (m) represents the change in the amount owed based on the monthly payments made.
Barton starts with a car loan of $2,625 and makes monthly payments of $175. Thus, after each payment, the remaining balance decreases by $175.
The slope can be calculated using the formula:
\[ m = \frac{\text{change in amount}}{\text{change in time}} \]
When Barton makes one payment, his balance goes from $2,625 to $2,625 - $175 = $2,450.
Let's evaluate each of the options provided to find which one corresponds to a decrease of $175 over a month (1 unit of time).
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\( m = \frac{1,050 - 1,225}{9 - 8} \)
- This represents a change of -175 over 1 month.
- This is a correct model.
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\( m = \frac{525 - 700}{11 - 12} \)
- This represents a change of 175 over -1 month, which is not valid for this context.
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\( m = \frac{2,450 - 2,625}{0 - 1} \)
- This would produce a slope that reflects an increase, not a decrease (change of -175 overall for a change of -1 month).
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\( m = \frac{9 - 8}{1,050 - 1,225} \)
- This represents a very small change in time relative to a large change in amount, and does not reflect Barton's loan repayment context.
Based on this analysis, the first option \( m = \frac{1,050 - 1,225}{9 - 8} \) is the only one that accurately reflects the situation, where the balance declines due to monthly payments of $175. Thus, the correct response is:
m = \frac{1,050 - 1,225}{9 - 8}