## Task 1 - 271221

Determine all pairs $(x, y)$ of non-zero real numbers $x, y$ that satisfy the following system of equations (1), (2):

$$
\begin{aligned}
& x \cdot\left(1+\frac{x}{y}\right)=6 \\
& y \cdot\left(1+\frac{y}{x}\right)=3
\end{aligned}
$$
The term $x+y$ is calculated from (1) and (2) as

$$
x+y=\frac{6 y}{x} \quad \text{or} \quad x+y=\frac{3 x}{y}
$$

from which, by equating, the equation

$$
\left(\frac{x}{y}\right)^{2}=2 \quad \text{or} \quad x= \pm \sqrt{2} y
$$

immediately follows. By substituting (*) into (1), one first obtains $x$ as

$$
x=\frac{6}{1+\frac{x}{y}}=\frac{6}{1 \pm \sqrt{2}}=6( \pm \sqrt{2}-1)
$$

and thus also $y$ as

$$
y= \pm \frac{x}{\sqrt{2}}=3(2 \mp \sqrt{2})
$$

i.e., each of the two sign choices in (*) yields a solution.