How to solve the following system of equations?

Question

$\left\{ \begin{aligned} xy + 2x + 2y &= -8\\ yz + 2y + 2z &= 24\\ xz + 2x + 2z &= -11 \end{aligned} \right.$ I need to solve it over the set of real numbers.

Answer 1

Note that $$(x+2)(y+2)=xy+2x+2y+4$$Hence, our system of equations becomes $$(x+2)(y+2)=-4$$$$(y+2)(z+2)=28$$$$(x+2)(z+2)=-7$$Can you solve from here?

Answer 2

Elimimating the variable $x$ from (1) and plug this in (3) we get $$-\frac{2 (y+4) z}{y+2}-\frac{4 (y+4)}{y+2}+2 z+11=0$$ Now we get $z$ from (2) $$z=-\frac{2 (y-12)}{y+2}$$ finally we obtain one equation in $y$ factorizing we get $$\frac{7 (y-2) (y+6)}{(y+2)^2}=0$$ From here we find the Solutions of our System.