6.243. $\left\{\begin{array}{l}5 \sqrt{x^{2}-3 y-88}+\sqrt{x+6 y}=19 \\ 3 \sqrt{x^{2}-3 y-88}=1+2 \sqrt{x+6 y} .\end{array}\right.$
## Solution.

Let $\left\{\begin{array}{l}\sqrt{x^{2}-3 y-88}=u \geq 0, \\ \sqrt{x+6 y}=v \geq 0 .\end{array}\right.$

With respect to $u$ and $v$, the system of equations has the form

$$
\left\{\begin{array}{l}
5 u+v=19, \\
3 u=1+2 v
\end{array} \Rightarrow u=\frac{1+2 v}{3}, 5\left(\frac{1+2 v}{3}\right)+v=19 \Rightarrow v=4, u=\frac{1+2 \cdot 4}{3}=3\right.
$$

Then $\left\{\begin{array}{l}\sqrt{x^{2}-3 y-88}=3 \\ \sqrt{x+6 y}=4\end{array} \Leftrightarrow\left\{\begin{array}{l}x^{2}-3 y-88=9, \\ x+6 y=16\end{array} \Rightarrow x=16-6 y\right.\right.$,

$$
(16-6 y)^{2}-3 y=97 \Leftrightarrow 36 y^{2}-195 y+159=0, \text { from which } y_{1}=1, y_{2}=\frac{53}{12}
$$

Then $x_{1}=10, x_{2}=-\frac{21}{2}$.

Answer: $(10 ; 1),\left(-\frac{21}{2} ; \frac{53}{12}\right)$.