Consider the following non-homogeneous system of 3 linear equations with 4 real unknowns, where $ \alpha \in \mathbb{R}$. See matrix augmented below.
\begin{pmatrix}\left(\alpha \:-3\right)&-2&\left(\alpha \:+10\right)&-1&19\\ \:9&5&-\left(\alpha \:+2\right)&-6&\left(3\alpha \:-10\right)\\ \:4&3&-\alpha \:&-5&-9\end{pmatrix}.
Determine the real $\alpha$ values for which the system is consistent and in this case find the general solutions as well as check for $\alpha$ for which the system is inconsistent.