To solve the problem, let's break down the scenario step by step.
1. Understanding the Problem:
- The 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.
2. Defining Variables:
- Let [tex]\( x \)[/tex] be the speed of the plane in miles per hour in still air.
- Let [tex]\( y \)[/tex] be the speed of the wind in miles per hour.
3. Using the Formula [tex]\( d = rt \)[/tex]:
With the Wind:
When the plane is flying with the wind, the effective speed is the sum of the plane's speed and the wind's speed ([tex]\( x + y \)[/tex]). The formula [tex]\( d = r \cdot t \)[/tex] translates to:
[tex]\[ 120 = (x + y) \cdot 0.75 \][/tex]
Rewriting this equation, we get:
[tex]\[ 0.75(x + y) = 120 \][/tex]
Against the Wind:
When the plane is flying against the wind, the effective speed is the plane's speed minus the wind's speed ([tex]\( x - y \)[/tex]). The formula [tex]\( d = r \cdot t \)[/tex] translates to:
[tex]\[ 120 = (x - y) \cdot 1 \][/tex]
Rewriting this equation, we get:
[tex]\[ x - y = 120 \][/tex]
4. Forming the System of Linear Equations:
The two essential equations we derived are:
[tex]\[ 0.75(x + y) = 120 \][/tex]
[tex]\[ x - y = 120 \][/tex]
These equations represent the system of linear equations in terms of the speed of the plane ([tex]\( x \)[/tex]) and the speed of the wind ([tex]\( y \)[/tex]).
So, the correct system of linear equations from the given choices is:
[tex]\[
\begin{array}{l}
0.75(x + y) = 120 \\
x - y = 120
\end{array}
\][/tex]
