Set of 6 equations with 6 unknowns

Question

Find all real solutions of
$\begin{cases}x=-t-z\\y=tz\\t=-u-q\\z=uq \\u=-x-y\\q=xy\end{cases}$

By multiplying all the sides of all the equations respectively we get that either one of the unknowns is equal to zero or that $xtu=1$. Nevertheless, I don’t know how to find solutions for any of these cases. Other methods I’ve tried lead me to expressions with an unknown raised to the 3rd power.
Thanks for your help in advance!

Answer 1

Using elimination, $q$ satisfies $q(q+2)(q^6-2q^5+4q^4-10q^3+16q^2-12q+4)=0$. The solution $q=0$ does not lead to any solutions. When $q=-2$ then $t=1, u=1, x=1, y=-2, z=-2$ is the only solution.