Unique solution through the determinant

For $a\in\mathbb{R}$, consider the system of linear equations $$\begin{aligned}a\,x+a\,y\qquad&=a+2\\x+a\,y+(a-1)z&=a-4\\a\,x+a\,y+(a-2)z&=-8\end{aligned}$$ in the unknowns $x$, $y$ and $z$. Then, which of the following statements is TRUE?