Let's approach this step-by-step to determine which system of equations can be used to find [tex]\( x \)[/tex], the speed of the boat in miles per hour, and [tex]\( y \)[/tex], the speed of the current in miles per hour.
1. Understand the problem:
- The boat travels downstream with the current.
- Distance traveled downstream: 9 miles.
- Time taken for the downstream journey: 90 minutes (which is 1.5 hours).
2. Define variables:
- Let [tex]\( x \)[/tex] be the speed of the boat in still water (miles per hour).
- Let [tex]\( y \)[/tex] be the speed of the current (miles per hour).
3. Determine the effective speed:
- When the boat is traveling downstream, its effective speed is [tex]\( (x + y) \)[/tex] because the current aids the boat.
4. Use the distance formula:
- The formula for distance [tex]\( d = r \cdot t \)[/tex], where [tex]\( r \)[/tex] is the rate (or speed) and [tex]\( t \)[/tex] is the time.
- Given [tex]\( d = 9 \)[/tex] miles and [tex]\( t = 1.5 \)[/tex] hours, we can set up the equation for downstream travel:
[tex]\[
9 = 1.5 \cdot (x + y)
\][/tex]
5. Formulate the equation:
- Rearrange the equation to solve for [tex]\( (x + y) \)[/tex]:
[tex]\[
x + y = \frac{9}{1.5}
\][/tex]
[tex]\[
x + y = 6
\][/tex]
- However, since we are comparing to options provided:
- The correct equation directly relating distance, speed, and time is:
[tex]\[
9 = 1.5 \cdot (x + y)
\][/tex]
From the given multiple-choice options, the correct one that matches our derived equation is:
[tex]\[
9 = 1.5(x + y)
\][/tex]
Hence, the correct answer is:
[tex]\[
9 = 1.5(x + y)
\][/tex]
