To determine the correct function for the loan balance \( y \) after \( x \) monthly payments, we can analyze the information given and create a linear equation.
- After 6 payments, the balance is \( y = 12,150 \).
- After 11 payments, the balance is \( y = 11,025 \).
First, we calculate the change in balance over the change in payments:
\[ \text{Change in balance} = 12,150 - 11,025 = 1,125 \]
\[ \text{Change in months} = 11 - 6 = 5 \]
Now we can determine the monthly payment amount:
\[ \text{Monthly payment} = \frac{1,125}{5} = 225 \]
So, the loan balance decreases by \( 225 \) for each month of payment.
Next, we can find the initial loan balance by recognizing that the balance after 6 payments is:
\[ y = \text{Initial balance} - 225 \times 6 \] Let’s denote the initial balance as \( B \). Therefore:
\[ 12,150 = B - 1,350 \quad \text{(since } 6 \times 225 = 1,350\text{)} \] \[ B = 12,150 + 1,350 = 13,500 \]
This means the initial loan balance is \( 13,500 \).
Now we will express \( y \) as a function of \( x \), the number of monthly payments:
\[ y = B - 225x \]
Substituting \( B = 13,500 \),
\[ y = 13,500 - 225x \]
This can be rewritten as:
\[ y = -225x + 13,500 \]
Now we can compare this function with the options given:
- \( y = -225x + 13500 \) (matches)
- \( y = 225x - 13500 \)
- \( y = -225x + 13275 \)
- \( y = 225x - 13275 \)
The correct function that can be used to find \( y \), the balance of the loan after \( x \) monthly payments, is:
\[ \boxed{y = -225x + 13500} \]
1 answer