Example 24. Solve the system

$$
\left\{\begin{array}{l}
(3 / 2)^{x-y}-(2 / 3)^{x-y}=65 / 36 \\
x y-x+y=118
\end{array}\right.
$$
Solution. Denoting $(3 / 2)^{x-y}=t$, the first equation of the given system can be written as

$$
t-\frac{1}{t}=\frac{65}{36}
$$

This equation is equivalent to the equation

$$
\frac{36 t^{2}-65 t-36}{36 t}=0
$$

i.e., the equation

$$
\frac{36(t-9 / 4)(t+4 / 9)}{36 t}=0
$$

Since $t>0$, the last equation has a unique root, which is the number $9 / 4$.

Thus, the first equation of the system is equivalent to the equation $(3 / 2)^{x-y}=(3 / 2)^{2}$, i.e., the equation $x-y=2$. Therefore, the original system is equivalent to the system

$$
\left\{\begin{aligned}
x-y & =2 \\
x y & =120
\end{aligned}\right.
$$

the solutions of which are two ordered pairs of numbers: $(12 ; 10)$ and $(-10 ;-12)$.

90

CH. 2. EQUATIONS WITH ONE UNKNOWN