To determine the correct slope that models the remaining balance of Bartonβs car loan, we need to understand that the slope (m) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
In this context, \(y\) represents the remaining balance and \(x\) represents the number of payments made.
Given the information:
- Initially, Barton's loan amount is $2,625.
- After one payment of $175, his remaining balance would be \(2,625 - 175 = 2,450\).
- After two payments of $175, his remaining balance would be \(2,450 - 175 = 2,275\), and so on.
Next, let's analyze each option to see if they provide a correct slope that reflects the decrease in loan balance per payment.
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Option 1: \( m = \frac{525 - 700}{11 - 12} \)
- This simplifies to \( m = \frac{-175}{-1} = 175 \), which does not correspond with the loan decrease, because the balance decreases not increases.
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Option 2: \( m = \frac{9 - 8}{1,050 - 1,225} \)
- This simplifies to \( m = \frac{1}{-175} \), which does not make sense in the context of remaining balances.
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Option 3: \( m = \frac{2,450 - 2,625}{0 - 1} \)
- This simplifies to \( m = \frac{-175}{-1} = 175 \), which correctly indicates an increase in balance, hence not the right model for a loan.
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Option 4: \( m = \frac{1,050 - 1,225}{9 - 8} \)
- This simplifies to \( m = \frac{-175}{1} = -175\). This indicates that with every payment, the loan decreases by $175, which correctly models the remaining balance of the loan.
Thus, the correct example that models the slope of the remaining balance after each payment is Option 4:
\[ m = \frac{1,050 - 1,225}{9 - 8} \]