Example 1. Solve the system of equations
$$
\left\{\begin{array}{l}
x(y+z-x)=39-2 x^{2}, \\
y(z+x-y)=52-2 y^{2}, \\
z(x+y-z)=78-2 z^{2} .
\end{array}\right.
$$
Solving by expanding the brackets and factoring, the original system of equations can be transformed into
\[
\left\{\begin{array}{l}
x(x+y+z)=39, \\
y(x+y+z)=52, \\
z(x+y+z)=78 .
\end{array}\right.
\]
By adding (1) + (2) + (3) and then taking the square root, we get
\[ 
x+y+z= \pm 13.
\]
Substituting (4) into (1), (2), and (3) respectively, we obtain
\[
\left\{\begin{array}{l}
x_{1}=3, \\
y_{1}=4, \\
z_{1}=6 ;
\end{array} \quad\left\{\begin{array}{l}
x_{2}=-3, \\
y_{2}=-4, \\
z_{2}=-6 .
\end{array}\right.\right.
\]