6.119. $\left\{\begin{array}{l}\sqrt{x}-\sqrt{y}=0.5 \sqrt{x y} \\ x+y=5 .\end{array}\right.$
Solution.

Domain of definition: $\left\{\begin{array}{l}x \geq 0 \\ y \geq 0 .\end{array}\right.$

Let $\left\{\begin{array}{l}\sqrt{x}=u, \\ \sqrt{y}=v,\end{array}\right.$ where $u \geq 0$ and $v \geq 0$.

With respect to $u$ and $v$, the system takes the form

$$
\begin{aligned}
& \left\{\begin{array} { l } 
{ u - v = 0 . 5 u v , } \\
{ u ^ { 2 } + v ^ { 2 } = 5 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
u-v=0.5 u v, \\
(u-v)^{2}+2 u v=5
\end{array} \Rightarrow\right.\right. \\
& \Rightarrow(0.5 u v)^{2}+2 u v-5=0, \quad 0.25(u v)^{2}+2 u v-5=0
\end{aligned}
$$

from which $u v=-10$ or $u v=2 ; u v=-10<0$ is not suitable. Then

$$
\left\{\begin{array} { l } 
{ u - v = 0 . 5 u v , } \\
{ u v = 2 }
\end{array} \Leftrightarrow \left\{\begin{array} { l } 
{ u - v = 1 , } \\
{ u v = 2 , }
\end{array} \text { from which } \left\{\begin{array}{l}
u=2, \\
v=1 ;
\end{array},\left\{\begin{array} { l } 
{ \sqrt { x } = 2 , } \\
{ \sqrt { y } = 1 , }
\end{array} \quad \left\{\begin{array}{l}
x=4 \\
y=1
\end{array}\right.\right.\right.\right.\right.
$$

Answer: $(4 ; 1)$.