Let's solve the problem step by step.
Let the rate of the plane in still air be [tex]\( p \)[/tex] (in kilometers per hour) and the rate of the wind be [tex]\( w \)[/tex] (in kilometers per hour).
### Step 1: Establish Equations
#### Against the Wind
When traveling against the wind, the effective speed of the plane is reduced by the speed of the wind. Thus, the effective speed of the plane is [tex]\( p - w \)[/tex].
The given data tells us that the plane travels 2650 kilometers in 5 hours while going against the wind. We can express this relationship with the equation:
[tex]\[ p - w = \frac{2650}{5} \][/tex]
Simplifying, we get:
[tex]\[ p - w = 530 \][/tex]
#### With the Wind
When traveling with the wind, the effective speed of the plane is increased by the speed of the wind. Thus, the effective speed of the plane is [tex]\( p + w \)[/tex].
The given data tells us that the plane travels 5700 kilometers in 6 hours while going with the wind. We can express this relationship with the equation:
[tex]\[ p + w = \frac{5700}{6} \][/tex]
Simplifying, we get:
[tex]\[ p + w = 950 \][/tex]
### Step 2: Solve the System of Equations
We now have a system of linear equations:
[tex]\[
\begin{cases}
p - w = 530 \\
p + w = 950
\end{cases}
\][/tex]
We can solve these equations simultaneously.
First, add the two equations:
[tex]\[
(p - w) + (p + w) = 530 + 950
\][/tex]
[tex]\[
2p = 1480
\][/tex]
[tex]\[
p = 740
\][/tex]
Next, substitute the value of [tex]\( p \)[/tex] back into one of the original equations to find [tex]\( w \)[/tex]. Let's use [tex]\( p + w = 950 \)[/tex]:
[tex]\[
740 + w = 950
\][/tex]
[tex]\[
w = 950 - 740
\][/tex]
[tex]\[
w = 210
\][/tex]
### Conclusion
The rate of the plane in still air is [tex]\( 740 \)[/tex] kilometers per hour, and the rate of the wind is [tex]\( 210 \)[/tex] kilometers per hour.
[tex]\[
\begin{tabular}{|c|c|}
\hline
Rate of the plane: & 740 km/h \\
\hline
Rate of the wind: & 210 km/h \\
\hline
\end{tabular}
\][/tex]
