To determine the correct system of equations for this problem, let's analyze the given information step by step.
1. Understanding the Problem:
- A motorboat travels 9 miles downstream in 30 minutes.
- It takes 90 minutes to travel the same distance upstream.
2. Define Variables:
- Let [tex]\( x \)[/tex] be the speed of the boat in still water (in miles per hour).
- Let [tex]\( y \)[/tex] be the speed of the current (in miles per hour).
3. Convert Time to Hours:
- Downstream travel time: 30 minutes [tex]\( = \frac{30}{60} \)[/tex] hours [tex]\( = 0.5 \)[/tex] hours.
- Upstream travel time: 90 minutes [tex]\( = \frac{90}{60} \)[/tex] hours [tex]\( = 1.5 \)[/tex] hours.
4. Recall the Distance Formula:
- Distance = Rate × Time ( [tex]\( d = r \cdot t \)[/tex] ).
5. Formulate Equations:
- Downstream: When the boat is traveling downstream, the speed of the boat and the speed of the current add up. Thus, the effective speed is [tex]\( (x + y) \)[/tex]. Using the distance formula:
[tex]\[
9 = (x + y) \cdot 0.5
\][/tex]
- Upstream: When the boat is traveling upstream, the boat’s speed is reduced by the current's speed. Thus, the effective speed is [tex]\( (x - y) \)[/tex]. Using the distance formula:
[tex]\[
9 = (x - y) \cdot 1.5
\][/tex]
6. Simplify the Equations:
- For the downstream equation:
[tex]\[
9 = 0.5(x + y)
\][/tex]
- For the upstream equation:
[tex]\[
9 = 1.5(x - y)
\][/tex]
Thus, the correct system of equations to find the speed of the boat [tex]\( x \)[/tex] and the speed of the current [tex]\( y \)[/tex] is:
[tex]\[
9 = 0.5(x + y) \quad \text{and} \quad 9 = 1.5(x - y)
\][/tex]
Therefore, the correct answer is:
[tex]\[
9 = 0.5(x + y) \quad \text{and} \quad 9 = 1.5(x - y)
\][/tex]
