Find all $x,y,z$ satisfying $xy=z-x-y$ and cyclic permutations

Question

Find all ordered pairs $(x,y,z)$ real numbers, which satisfy the following system of equations: $$xy=z-x-y\\xz=y-x-z\\yz=x-y-z$$

Answer 1

Hint: $$xy=z-x-y \quad \iff \quad (x+1)(y+1)=z+1.$$

Hence $(x+1)(y+1)=z+1$, $(x+1)(z+1)=y+1$, $(z+1)(y+1)=x+1$.

Answer 2

Another hint: Subtracting the third equation from the second reveals a constant value for z as long as x ≠ y (Thank you Calvin).

$z(x-y)=-2(x-y)$

Answer 3

The groebner basis in lex order is (try sympy)

$$x - z^4 - z^3 + 3z^2 + 2z=0$$ $$y + z^4 + 2z^3 - z^2 - 3z=0$$ $$z^5 + 3z^4 - 6z^2 - 4z=0$$

Note that now the first equation only contains $x,z$, the second only $y,z$ and the third only $z$.

If you factor out one $z$ from the third equation you have a degree four polynomial that is solvable in formulas (or you carry on guessing and factoring for that polynomial).