To determine how much Harry pays back to the bank each month, we need to analyze the given equation that represents the remaining loan amount:
[tex]\[
D = -200t + 9000
\][/tex]
Here, [tex]\(D\)[/tex] is the remaining amount of the loan after [tex]\(t\)[/tex] months. We aim to find out the amount paid back each month.
### Step-by-Step Breakdown
1. Identify the Terms in the Equation:
- The equation [tex]\(D = -200t + 9000\)[/tex] is in the form of a linear equation [tex]\(D = mt + b\)[/tex], where:
- [tex]\(m\)[/tex] represents the rate of change of the remaining loan amount per month.
- [tex]\(b\)[/tex] represents the initial loan amount when [tex]\(t\)[/tex] is 0.
2. Interpret the Coefficient of [tex]\(t\)[/tex]:
- The coefficient of [tex]\(t\)[/tex] in this equation is [tex]\(-200\)[/tex]. This coefficient indicates how much the remaining loan amount (D) decreases each month. A negative sign indicates a decrease.
3. Monthly Payment:
- Since the remaining loan decreases by [tex]\(-200\)[/tex] dollars each month, this means Harry pays back [tex]\(200\)[/tex] dollars every month (the negative sign signifies the reduction in debt, not a negative payment).
### Conclusion
From the breakdown of the equation [tex]\(D = -200t + 9000\)[/tex], we find that Harry pays back the following amount each month:
[tex]\[
\boxed{200}
\][/tex]
Thus, Harry pays back [tex]\(200\)[/tex] dollars to the bank each month.
