Question

an aircraft travels with the wind for 120 miles in 0.75 of an hour. the return trip is flown against the wind and takes exactly 1 hour.\
which system of linear equations represents ( x ), the speed of the plane in miles per hour, and ( y ), the speed of the wind in miles per hour? recall the formula ( d = rt ).\
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$$\begin{cases} 120(x - y) = 0.75 \\\\ 120(x + y) = 1 \\end{cases}$$

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$$\begin{cases} 120(x + y) = 0.75 \\\\ 120(x - y) = 1 \\end{cases}$$

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$$\begin{cases} 0.75(x + y) = 120 \\\\ x - y = 120 \\end{cases}$$

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$$\begin{cases} 0.75(x - y) = 120 \\\\ x + y = 120 \\end{cases}$$

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Explanation:

Step1: Analyze with - wind trip

When the aircraft travels with the wind, its effective speed is the sum of the plane's speed (\(x\)) and the wind's speed (\(y\)), so the speed is \(x + y\). The distance \(d = 120\) miles and time \(t = 0.75\) hours. Using the formula \(d=rt\), we substitute the values: \(120 = 0.75(x + y)\), which can be rewritten as \(0.75(x + y)=120\).

Step2: Analyze against - wind trip

When the aircraft travels against the wind, its effective speed is the difference between the plane's speed (\(x\)) and the wind's speed (\(y\)), so the speed is \(x - y\). The distance \(d = 120\) miles and time \(t = 1\) hour. Using the formula \(d = rt\), we substitute the values: \(120=1\times(x - y)\), which is \(x - y = 120\)? Wait, no, wait. Wait, the time for against wind is 1 hour, distance is 120, so \(d=rt\) gives \(120=(x - y)\times1\), so \(x - y=120\)? Wait, no, no, I made a mistake. Wait, the third option is \(0.75(x + y)=120\) and \(x - y = 120\)? Wait, no, let's re - check.

Wait, when going with the wind: speed is \(x + y\), time is \(0.75\) hours, distance is \(120\) miles. So \(d=rt\) implies \(120=(x + y)\times0.75\), which is \(0.75(x + y)=120\).

When going against the wind: speed is \(x - y\), time is \(1\) hour, distance is \(120\) miles. So \(d = rt\) implies \(120=(x - y)\times1\), which is \(x - y=120\)? Wait, no, that can't be. Wait, no, the third option is \(0.75(x + y)=120\) and \(x - y = 120\)? Wait, no, let's check the options again.

Wait, the third option is:

\(0.75(x + y)=120\)

\(x - y = 120\)

Wait, no, when against the wind, time is 1 hour, distance is 120, so \(r=\frac{d}{t}=\frac{120}{1}=120\), and \(r=x - y\), so \(x - y = 120\). And with the wind, \(r=x + y\), \(t = 0.75\), \(d = 120\), so \(x + y=\frac{120}{0.75}=160\), and \(0.75(x + y)=120\) is correct. So the third option is the correct system.

Answer:

\(0.75(x + y)=120\)

\(x - y = 120\) (the third option among the given boxes)