To find the correct slope that models Barton’s car loan after making monthly payments, we need to interpret the meaning of the slope in this context.
The slope of a line in a graph represents the rate of change of the remaining balance after each payment. Since Barton is making monthly payments of $175, the slope should reflect this decrease in balance for each month.
The general formula for slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Where:
- \(y_2\) and \(y_1\) are the remaining balances after different payments (in dollars).
- \(x_2\) and \(x_1\) are the corresponding months for which those balances occur.
In the context, if we analyze the proposed options:
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\(m=\frac{9-8}{1,050-1,225}\): This does not represent a realistic change in the remaining balance since the numerator is a change in the month and the denominator represents a change in balance, which is not related properly.
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\(m=\frac{525-700}{11-12}\): This suggests illogical values since it implies an increase in balance which doesn't fit.
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\(m=\frac{1,050-1,225}{9-8}\): This also suggests a change that correlates months with remaining balance but might represent an increase rather than a decrease.
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\(m=\frac{2,450-2,625}{0-1}\): This is ideal as it directly links the loan balance and time, showing a clear reduction of the balance of $175 per month as it reflects a full payment period.
Since Barton is making consistent monthly payments of $175, a more correct understanding should lead us towards finding the slope that would account for his consistent monthly payment reduction.
The only option that reasonably fits is given by the notion of making a payment (leading to a decrease in balance). Therefore, this indicates the slope would be a negative value consistent annually adjusted by the payments made which would also be financially numerically meaningful.
In conclusion, while none of the given responses directly calculate the slope of the remaining balance based on the monthly payment model, if forced to choose based on available options, we would seek the expression that provides a consistent and logical relationship between months and remaining loan balance.
However, the correct slope representation for Barton making payments of $175 a month should ideally be depicted as a negative reduction in his balance. Thus, he would be paying off the balance as monthly slopes illustrate decreases toward loan payoff, which would be modeled on values representing lower remaining balances against the month states he pays them.