3. Find all triples of pairwise distinct real numbers $x, y, z$ that are solutions to the system of equations:

$$
\left\{\begin{array}{l}
x^{2}+y^{2}=-x+3 y+z \\
y^{2}+z^{2}=x+3 y-z \\
z^{2}+x^{2}=2 x+2 y-z
\end{array}\right.
$$
3. Find all triples of pairwise distinct real numbers $x, y, z$ that are solutions to the system of equations:

$$
\left\{\begin{array}{l}
x^{2}+y^{2}=-x+3 y+z \\
y^{2}+z^{2}=x+3 y-z \\
z^{2}+x^{2}=2 x+2 y-z
\end{array}\right.
$$

ANSWER: $x=0, y=1, z=-2$ or $x=-\frac{3}{2}, y=\frac{5}{2}, z=-\frac{1}{2}$.

SOLUTION. The difference of the first two equations gives:

$$
x^{2}-z^{2}=-2(x-z) \Rightarrow x+z=-2
$$

since the numbers are pairwise distinct by condition. The difference of the second and third equations gives:

$$
y^{2}-x^{2}=y-x \Rightarrow y+x=1
$$

Substituting the obtained relations $y=1-x$ and $z=-x-2$ into the first equation:

$$
x^{2}+(1-x)^{2}=-x+3(1-x)+(-x-2) \Rightarrow 2 x^{2}+3 x=0
$$

For $x=0, y=1, z=-2$. For $x=-\frac{3}{2}, y=\frac{5}{2}, z=-\frac{1}{2}$.