1. Solve the system of equations $\left\{\begin{array}{l}\sqrt{11 x-y}-\sqrt{y-x}=1, \\ 7 \sqrt{y-x}+6 y-26 x=3\end{array}\right.$
Solution. Introduce new variables $u=\sqrt{11 x-y}$ and $v=\sqrt{y-x}, u \geq 0, v \geq 0$. Then

$$
\left\{\begin{array} { l } 
{ 1 1 x - y = u ^ { 2 } , } \\
{ y - x = v ^ { 2 } }
\end{array} \left\{\begin{array}{l}
x=0.1\left(u^{2}+v^{2}\right) \\
y=0.1\left(u^{2}+11 v^{2}\right)
\end{array}\right.\right.
$$

And, moreover, $6 y-26 x=-2(11 x-y)+4(y-x)=-2 u^{2}+4 v^{2}$. Now, in the new variables, the system will take the form

$$
\left\{\begin{array} { l } 
{ u \geq 0 , v \geq 0 , } \\
{ u - v = 1 , } \\
{ 7 v + 4 v ^ { 2 } - 2 u ^ { 2 } = 3 , }
\end{array} \left\{\begin{array}{l}
u \geq 0, v \geq 0 \\
u=v+1 \\
2 v^{2}+3 v-5=0
\end{array}\right.\right.
$$

From the last equation of the system, we get $v=\frac{-3 \pm 7}{4}$, i.e. $\left\{\begin{array}{l}v=1, \\ u=2,\end{array}\right.$ hence, $\left\{\begin{array}{l}x=\frac{1}{2} \\ y=\frac{3}{2}\end{array}\right.$.

Answer: $(0.5 ; 1.5)$.