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

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

By bringing to a common denominator, we get

$$
\left\{\begin{array}{l}
6+x y=9 x, \\
x y+6=3 y
\end{array} \Rightarrow 9 x-3 y=0\right.
$$

from which $y=3 x$. From this, we get $\frac{2}{x}+\frac{3 x}{3}=3, x^{2}-3 x+2=0$ for $x \neq 0$, from which $x_{1}=1, x_{2}=2$; then $y_{1}=3, y_{2}=6$.

Answer: $(1 ; 3)(2 ; 6)$