7.282. $\left\{\begin{array}{l}\lg (x-3)-\lg (5-y)=0, \\ 4^{-1} \cdot \sqrt[y]{4^{x}}-8 \sqrt[x]{8^{y}}=0 .\end{array}\right.$
Solution.

Domain of definition: $\left\{\begin{array}{l}x>3 \\ 0 \neq y<5 .\end{array}\right.$

Rewrite the system of equations as

$\left\{\begin{array}{l}\lg \frac{x-3}{5-y}=0, \\ 2^{-2+\frac{2 x}{y}}=2^{3+\frac{3 y}{x}}\end{array} \Leftrightarrow\left\{\begin{array}{l}\frac{x-3}{5-y}=1, \\ \frac{2 x}{y}-2=3+\frac{3 y}{x}\end{array} \Rightarrow x=8-y\right.\right.$,
$\frac{2(8-y)}{y}-5-\frac{3 y}{8-y}=0 \Leftrightarrow y^{2}-18 y+32=0, y_{1}=2, y_{2}=16$.

Then $x_{1}=6, x_{2}=-8$.

$\left\{\begin{array}{l}x_{2}=-8, \\ y_{2}=16\end{array}\right.$ does not satisfy the domain of definition.

Answer: $(6 ; 2)$.