## Task 2 - 291212

Determine all pairs $(x, y)$ of real numbers that satisfy the following system of equations:

$$
\begin{aligned}
x^{3}+y^{3} & =7 \\
x+x y+y & =-1
\end{aligned}
$$
Let $x, y$ be real numbers that satisfy the system of equations (1), (2). Then, after rearranging equation (2),

$$
(x+1)(y+1)=0
$$

This equation holds if and only if $x=-1$ or $y=-1$.

Let $x=-1$. From (1) it follows that $y^{3}=8$, thus $y=2$.

Let $y=-1$. Then, from (1) it follows analogously that $y=2$.

Therefore, the pairs $(-1,2),(2,-1)$ can at most satisfy equations (1), (2). As a check shows, these two pairs are indeed solutions.

## Adopted from $[5]$