To solve: the given problem by modeling and solving a system of linear equation.

Procedure used:
"Model and solve system of linear equations"
1) Identify the category of given problem.
2) Recognize and algebraically name the unknowns.
3) Translate the problem statement as a system of linear equations using the variables from Step 2.
4) Solve the system as obtained in Step 3.
5) Check for the correctness of the solution by substituting back the values from Step 4 in the equation in Step 3 thereby verifying the statements in the given problem.
6) State the answer statement.
Calculation:
Step 1:
The given problem is a uniform motion problem where speed of the plane in still air and speed of wind are to be calculated using the following formula:
${\displaystyle time=\frac{dis\tan ce}{speed}}$
Step 2:
There are two unknowns in the problem. Let w the speed of wind and 150 mph is the speed of aircraft.
Step 3:
When aircraft flies into the wind, the speed is lowered by wind. Therefore, the average speed of the plane will be ${\displaystyle 150-w}$. In the case of aircraft with the wind at his back, so the speed will be ${\displaystyle 150+w}$, as summarized in Table 1.
$\begin{array}{|ccc|}\hline & Time& Speed\\ \text{With the wind}& 3hours& 150-w\\ \text{Against the wind}& 2hours& 150+w\\ \hline\end{array}$
Table 1
Distance travelled by aircraft into the wind and with the wind is equal.
Thus, the sytem of linear equation is:
${\displaystyle 2(150+w)=3(150-w)\left(1\right)}$
Step 4:
Solve the system of equations for w.
${\displaystyle 2(150+w)=3(150-w)}$
${\displaystyle 300+2w=450-3w}$
${\displaystyle 2w+3w=450-300}$
${\displaystyle 5w=150}$
Solve the equation above to obtain the value of variable w as:
${\displaystyle w=\frac{150}{5}}$
${\displaystyle =30}$
Thus, the value of ${\displaystyle w=30}$.
Step 5:
To check the solution obtained in Step 4, when aircraft flies against the wind, the distance travelled is ${\displaystyle 3(150-30)=360}$ miles and when plane flies with the wind, the distance travelled is ${\displaystyle 2(150+30)=360}$ miles.
Step 6:
Hence, the speed at which plane flies in still air is 150 mph. Also, the speed is reduced by 30 mph when plane flies into the wind and is increased by 30 mph and when plane flies with the wind at his back.