Question

a motorboat travels 9 miles downstream (with the current) in 30 minutes. the return trip upstream (against the current) takes 90 minutes.\
which system of equations can be used to find ( x ), the speed of the boat in miles per hour, and ( y ), the speed of the current in miles per hour? recall the formula ( d = rt ).\
\\(\

$$\begin{array}{|c|} \\hline 9 = 0.5(x - y) \\\\ 9 = 1.5(x + y) \\\\ \\hline \\end{array}$$

\\) \\(\

$$\begin{array}{|c|} \\hline 9 = 1.5(x - y) \\\\ 9 = 0.5(x + y) \\\\ \\hline \\end{array}$$

\\) \\(\

$$\begin{array}{|c|} \\hline 0.5 = 9(x - y) \\\\ 1.5 = 9(x + y) \\\\ \\hline \\end{array}$$

\\) \\(\

$$\begin{array}{|c|} \\hline 1.5 = 9(x - y) \\\\ 0.5 = 9(x + y) \\\\ \\hline \\end{array}$$

\\)

Explanation:

Step1: Convert time to hours

Downstream time: 30 minutes = $\frac{30}{60}$ = 0.5 hours.
Upstream time: 90 minutes = $\frac{90}{60}$ = 1.5 hours.

Step2: Determine downstream and upstream rates

Downstream rate (with current): \( x + y \) (boat speed + current speed).
Using \( d = rt \), downstream: \( 9 = 0.5(x + y) \).

Upstream rate (against current): \( x - y \) (boat speed - current speed).
Using \( d = rt \), upstream: \( 9 = 1.5(x - y) \).

Answer:

The system of equations is \(\boldsymbol{

$$\begin{cases} 9 = 1.5(x - y) \\ 9 = 0.5(x + y) \end{cases}$$

}\) (the second option: \( 9 = 1.5(x - y) \); \( 9 = 0.5(x + y) \))