Several functions in Mathematica can do it. Example: use the Reduce function:
Reduce[{x + 2 y - 3 z == 4 2 x - y + 5 z == 2 4 x + 3 y + a^2 z == a + 3}, {x, y, z}]
From the result you should always get the linear system solvable for
$a \in \mathbb{R}$
