Generally speaking, you should NEVER compute the determinant of anything. The determinant is essentially a theoretical tool. It's almost never the best thing to compute, even when it is actually equivalent to the questions being asked, and in this case it isn't even "morally" the right thing to do. By that, I mean the question (in matrix form) is literally asking when a fixed vector is in the span of 3 other vectors. I can't imagine anything more absurd than computing the determinant of a matrix to determine this. Not only is this far more complicated than simple row reduction, but it doesn't even properly answer the question, since the equations can be consistent even if the determinant is 0 e.g. if the right-hand side is in the span on the columns for any choice of $A$. Instead, you just row reduce. For this example, you do the usual trick of augmenting the right hand side so that all row operations are carried out automatically. This gives you the matrix

\[

\begin{pmatrix}

3 & 1 & -2 & -7 \\

1 & 1 & a & -6 \\

2 & 2 & 1 & 9

\end{pmatrix}

\]

After row reducing, you are left with the equivalent matrix

\[

\begin{pmatrix}

3 & 1 & -2 & -7 \\

0 & \frac{2}{3} & a + \frac{2}{3} & -\frac{11}{3} \\

0 & 0 & 1 - 2a & 21

\end{pmatrix}

\]

From this matrix, it is trivial to see that all 3 equations are consistent if and only if the last equation is consistent. The last equation reads $(1-2a)z = 21$, which is consistent provided $1-2a \neq 0$.