6.070. $\left\{\begin{array}{l}\frac{x}{y}+\frac{y}{x}=\frac{13}{6}, \\ x+y=5\end{array}\right.$
Solution.

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

Multiplying both sides of the first equation by $6 x y \neq 0$, we get

$$
\begin{aligned}
& \left\{\begin{array} { l } 
{ 6 ( x ^ { 2 } + y ^ { 2 } ) = 1 3 x y , } \\
{ x + y = 5 }
\end{array} \Leftrightarrow \left\{\begin{array} { l } 
{ 6 ( ( x + y ) ^ { 2 } - 2 x y ) = 1 3 x y , } \\
{ x + y = 5 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
x y=6 \\
x+y=5,
\end{array}\right.\right.\right. \\
& \left\{\begin{array}{l}
y^{2}-5 y+6=0, \\
x+y=5,
\end{array},\left\{\begin{array}{l}
x_{1}=2, \\
y_{1}=3,
\end{array},\left\{\begin{array}{l}
x_{2}=3, \\
y_{2}=2 .
\end{array}\right.\right.\right.
\end{aligned}
$$

Answer: $(2 ; 3),(3 ; 2)$.