## Task 1 - 221221

Determine all triples $(x, y, z)$ of real numbers that satisfy the system of equations

$$
\begin{aligned}
& x \cdot(y+z)=5 \\
& y \cdot(x+z)=8 \\
& z \cdot(x+y)=9
\end{aligned}
$$
If we set

$$
u:=x y, v:=x z, w:=y z
$$

we obtain the linear(!) system of equations

$$
u+v=5, \quad u+w=8, \quad v+w=9
$$

in $u, v, w$ with the obvious solutions

$$
u=x y=2, \quad v=x z=3, \quad w=y z=6
$$

In particular, it follows that

$$
(x y z)^{2}=u v w=36 \Rightarrow x y z= \pm 6
$$

and thus

$$
x=\frac{x y z}{y z}= \pm 1, \quad y=\frac{x y z}{x z}= \pm 2, \quad z=\frac{x y z}{x y}= \pm 3
$$

Since due to $\left({ }^{*}\right)$ all variables must have the same sign, there are exactly the 2 solutions here

$$
(x, y, z) \in\{(1,2,3),(-1,-2,-3)\}
$$