exponential equation system without log

Question

How should I solve this equation system without using logarythms,using just a simple method? (E.g. turning it into a quadratic one using t)

$$\left(\frac{3}{2}\right)^{x-y} - \left(\frac{2}{3}\right)^{x-y}=\left(\frac{65}{36}\right)\\ xy-x+y=118$$

Answer 1

Hint : Use $$\frac{2}{3}^{x-y}=\frac{1}{\frac{3}{2}^{x-y}}$$ and solve $$t-\frac{1}{t}=\frac{65}{36}$$ The solutions are $t=\frac{9}{4}$ and $t=-\frac{4}{9}$.

Now, determine which t works if you have $\frac{3}{2}^{x-y}=t$

Finally you get $x-y=2$ together with the second equation.

The rest should be easy. Finally, you get $x=12$ and $y=10$ as one solution and $x=-10$ and $y=-12$ as the other solution.