Find all solutions, in the set of positive real numbers, of the system of equations:

$$
\left\{\begin{array}{l}
x(x+y+z)=26 \\
y(x+y+z)=27 \\
z(x+y+z)=28
\end{array}\right.
$$
Solution

By adding the three equations, we get $(x+y+z)^{2}=81$, that is, $x+y+z=9$, since we want positive solutions. Substituting this value into each equation, we have: $x=26 / 9, y=27 / 9=3$ and $z=28 / 9$. Thus, the only solution to the system is $(x, y, z)=(27 / 9,3,28 / 9)$.