Prove that if $3xy + 2yz + z + 1 = 0$ and $3zx + 2z + x + 1 = 0$, then $3xy + 2x + y + 1 = 0$.
I think the solution will involve combining these two equations in some way. I have attempted many possible combinations, such as multiplying i) by $x$, multiplying ii) by $y$ and then subtracting them, but there are always terms which do not cancel out, such as $3x^2y$ in this case.
This is false. Take $$ (x,y,z)=(2/3,-1/2,-5/12). $$ Then the first two equations are satisfied, but not the third one.