Solve in nonzero real numbers the system of equations $\{\begin{array}{l}\frac{7}{x}-\frac{27}{y}=2x^2\\ \frac{9}{y}-\frac{21}{x}=2y^2\end{array}$ Find the value of $\text{x }–\text{ y?}$

First, we will consider the system as a system with the unknowns $\frac{1}{x}\text{and}\frac{1}{y}$ . This is a linear system, so rather easy to solve. We obtain

$\frac{-28}{x}=x^2+3y^2\text{and}\frac{-36}{y}=3x^2+y^2\text{.}$

Multiplying the first equation by x and the second one by y we obtain the equivalent system
$\{\begin{array}{l}{x}^3+3x{y}^2=-28\\ 3x^{2}y+y^3=-36.\end{array}$
We recognize the terms of the expansion of  $(x+y{)}^3\text{and}(x-y{)}^3$ By adding and subtracting the equations, we obtain another form of the system
$\{\begin{array}{l}(x+y{)}^3=-64\\ (x-y{)}^3=8\end{array}$

This is quite easy to solve, since it is equivalent to $x+y=-4 and x-y=2,$ then $x=-1 and y=-3.$