## Task 2 - 231212

Determine all triples $(x ; y ; z)$ of non-zero real numbers that satisfy the following system of equations (1), (2), (3):

$$
\begin{aligned}
& x+\frac{1}{y}=-2 \\
& y+\frac{1}{z}=-\frac{1}{2} \\
& \frac{1}{x}+z=1
\end{aligned}
$$
If $(x ; y ; z)$ is a triple with the required properties, then due to (1) we have

$$
x b=-2-\frac{1}{y}=-\frac{2 y+1}{y}
$$

From this and $x \neq 0$ it follows that

$$
\frac{1}{x}=-\frac{y}{2 y+1}
$$

Due to (2), $\frac{1}{z}=-\frac{1}{2}-y=-\frac{2 y+1}{2}$, thus

$$
z=-\frac{2}{2 y+1}
$$

Substituting (4) and (5) into (3), we get

$$
-\frac{y}{2 y+1}-\frac{2}{2 y+1}=1 \quad \Rightarrow \quad y=-1
$$

and thus further

$$
x=-1 \quad ; \quad z=2
$$

Therefore, only the triple $(x ; y ; z)=(-1 ;-1 ; 2)$ can have the required properties. It does have these properties, as a check shows.

Adapted from $[5]$