= Solution
The third left-hand side is three times the first minus the second. Consistency of this <linear system> therefore requires $2=3a-a^2$, or
$$
a^2-3a+2=(a-1)(a-2)=0.
$$
This is also sufficient: the first two rows are <linearly independent>, and on setting the free variable $x_3=t$ they give
$$
\boxed{a\in\{1,2\},\qquad (x_1,x_2,x_3)=(a-t,\ a^2-2a-3t,\ t),\quad t\in\mathbb R.}
$$
Substitution into the third equation gives $3a-a^2=2$ for either allowed $a$. Explicitly the solution families are $(1-t,-1-3t,t)$ and $(2-t,-3t,t)$. For every other $a$, the same row relation gives an immediate inconsistency.
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