7.284. $\left\{\begin{array}{l}x+y=12, \\ 2\left(2 \log _{y^{2}} x-\log _{1 / x} y\right)=5 .\end{array}\right.$
Solution.

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

In the second equation of the system, we switch to base $y$. We have $2\left(\log _{y} x+\frac{1}{\log _{y} x}\right)=5 \Leftrightarrow 2 \log _{y}^{2} x-5 \log _{y} x+2=0 \Rightarrow \log _{y} x=\frac{1}{2}$ or $\log _{y} x=2$. Further, we get:

1) $\log _{y} x=2 \Leftrightarrow x=y^{2}$. From the first equation of the system $y^{2}+y-12=0 \Rightarrow$ $\Rightarrow y_{1}=3, y_{2}=-4$ does not fit the domain of definition, $x_{1}=9$;
2) $\log _{y} x=\frac{1}{2} \Leftrightarrow y=x^{2}$. From the first equation of the system $x^{2}+x-12=0 \Rightarrow$ $\Rightarrow x_{3}=3, x_{4}=-4$ does not fit the domain of definition, $y_{3}=9$.

Answer: $(3 ; 9),(9 ; 3)$.