7.291. $\left\{\begin{array}{l}\lg (x+y)-\lg 5=\lg x+\lg y-\lg 6, \\ \frac{\lg x}{\lg (y+6)-(\lg y+\lg 6)}=-1 .\end{array}\right.$
Solution.

Domain of definition: $\left\{\begin{array}{l}x>0, \\ y>0, \\ y \neq \frac{6}{5}, \\ y>-6 .\end{array}\right.$

From the condition we have

$$
\begin{aligned}
& \left\{\begin{array} { l } 
{ \operatorname { l g } \frac { x + y } { 5 } = \operatorname { l g } \frac { x y } { 6 } , } \\
{ \operatorname { l g } x = \operatorname { l g } \frac { 6 y } { y + 6 } }
\end{array} \Leftrightarrow \left\{\begin{array} { l } 
{ \frac { x + y } { 5 } = \frac { x y } { 6 } , } \\
{ x = \frac { 6 y } { y + 6 } }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
\frac{6 y}{y+6}+y \\
5=\frac{6 y}{y+6}
\end{array}=\frac{y^{2}}{y+6},\right.\right.\right. \\
& \Leftrightarrow\left\{\begin{array}{l}
y=3, \\
x=\frac{6 \cdot 3}{3+6}=2 .
\end{array}\right.
\end{aligned}
$$

Answer: $(2 ; 3)$.