3 unknown, 3 nonlinear equations of form $xy - z =$ constant

Question

How can I solve: $$xy - z = a $$ $$xz - y = b $$ $$ yz - x = c $$ for $x, y, z$ (where $a,b,c$ are constants)? Let all variables and constants be integers (or at least rational)

Answer 1

The brute force method would be to isolate (“solve for”) $z$ in the third equation (assuming $y \neq 0$), and substitute into the first two. That gives two nonlinear equations in two variables $x$ and $y$. Then repeat: isolate either $y$ or $x$ in one of those new equations, and substitute into the other. That should result in one (probably nonlinear) equation for one of the variables.