7.143. $\left\{\begin{array}{l}2^{\frac{x+y}{3}}+2^{\frac{x+y}{6}}=6 \\ x^{2}+5 y^{2}=6 x y\end{array}\right.$

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7.143. $\left\{\begin{array}{l}2^{\frac{x+y}{3}}+2^{\frac{x+y}{6}}=6 \\ x^{2}+5 y^{2}=6 x y\end{array}\right.$
## Solution.

From the condition $\left(2^{\frac{x+y}{6}}\right)^{2}+2^{\frac{x+y}{6}}-6=0$. Solving this equation as a quadratic equation in terms of $2^{\frac{x+y}{6}}$, we have $2^{\frac{x+y}{6}}=-3, \varnothing$; or $2^{\frac{x+y}{6}}=2$, from which $\frac{x+y}{6}=1, x+y=6$.

From the second equation of the system $x^{2}-6 y x+5 y^{2}=0$, solving it as a quadratic equation in terms of $x$, we have $x_{1}=y, x_{2}=5 y$. The original system is equivalent to two systems:

1) $\left\{\begin{array}{l}x+y=6 \\ x=y\end{array}\right.$
2) $\left\{\begin{array}{l}x+y=6, \\ x=5 y\end{array} \Rightarrow 1\right)\left\{\begin{array}{l}x_{1}=3 \\ y_{1}=3\end{array}\right.$
3) $\left\{\begin{array}{l}x_{2}=5 \\ y_{2}=1\end{array}\right.$

Answer: $\quad(3 ; 3),(5 ; 1)$