Let the speed of the boat be \(\mathrm{x} \mathrm{km} / \mathrm{hr}\) and the speed of the stream be \(\mathrm{y} \mathrm{kn} / \mathrm{hr}\). According to the question,
\(\begin{aligned} \frac{32}{x-y}+\frac{36}{x+y} &=7 \\ \text { and } \quad \frac{40}{x-y}+\frac{48}{x+y} &=9 \\ \text { Let } \frac{1}{x-y}=A, \frac{1}{x+y} &=B \\ \text { and } \quad 32 A+36 B &=7 \\ 40 A+48 B &=9 \quad \text {...(i) } \end{aligned}\)
Let \(\frac{1}{x-y}=A, \frac{1}{x+y}=B\),
and \(\quad \begin{array}{ll}32 A+36 B=7 & \text {...(i) } \\ 40 A+48 B=9 & \text {..(ii) }\end{array}\)
$$
\begin{aligned}
160 A+180 B &=35 \\
160 A+192 B &=36 \\
-\quad-&-1 \\
-12 B &=-1 \\
B &=\frac{1}{12}
\end{aligned}
$$
Substitute the value of \(B\) in (ii).Multiply equation (i) by 40 and (ii) by 32 and subtracting.
\(B\) in
$$
\begin{aligned}
40 A+48\left(\frac{1}{12}\right) &=9 \\
40 A+4 &=9 \\
40 A &=5 \\
A &=\frac{1}{8} \\
A &=\frac{1}{8} \text { and } B=\frac{1}{12}
\end{aligned}
$$
\(\Rightarrow\) \(\Rightarrow\)
then equations (iii) and (iv) become
$$
\begin{array}{r}
6 a-2 b=5 \\
2 a+6 b=5
\end{array}
$$
Multiplying eqn. (v) by 3 and then adding with eqn. (vi)),
$$
\begin{aligned}
20 a &=20 \\
a &=1
\end{aligned}
$$
Substituting this value of \(a\) in eqn. (v),
$$
b=\frac{1}{2}
$$
Now
$$
\frac{1}{y}=a=1
$$
or,
$$
\begin{aligned}
y &=1 \\
\frac{1}{x} &=b=\frac{1}{2}
\end{aligned}
$$
and
or, \(x=2\).
Hence, \(x=2, y=1\)
