7.280. $\left\{\begin{array}{l}2\left(\log _{1 / y} x-2 \log _{x^{2}} y\right)+5=0, \\ x y^{2}=32 .\end{array}\right.$
## Solution.

Domain of definition: $\left\{\begin{array}{l}0<x \neq 1, \\ 0<y \neq 1 .\end{array}\right.$

In the first equation of the system, we switch to base 2, and in the second equation, we take the logarithm to base 2. We have

$\left\{\begin{array}{l}2\left(-\frac{\log _{2} x}{\log _{2} y}-\frac{\log _{2} y}{\log _{2} x}\right)+5=0, \\ \log _{2} x+2 \log _{2} y=5\end{array} \Rightarrow \log _{2} x=5-2 \log _{2} y\right.$,

$2\left(\frac{2 \log _{2} y-5}{\log _{2} y}+\frac{\log _{2} y}{2 \log _{2} y-5}\right)+5=0 \Leftrightarrow 4 \log _{2}^{2} y-13 \log _{2} y+10=0$.

Solving the equation as a quadratic in $\log _{2} y$, we find

$\log _{2} y=\frac{5}{4}, y_{1}=2^{5 / 4}$, or $\log _{2} y=2, y_{2}=4$.

Then $\log _{2} x=\frac{5}{2}, x_{1}=2^{5 / 2} ; \log _{2} x=1, x_{2}=2$.

Answer: $(2 ; 4),(4 \sqrt{2} ; 2 \sqrt[4]{2})$.