# Determine the real $\alpha$ values ​for which the system is consistent. Find the general solutions.

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.

• HINT: Here you just need to use Gauss elimination. – MOVZBL Jun 19 '17 at 18:24
• So i found $\begin{pmatrix}\left(α-3\right)&-2&\left(α+10\right)&-1&19\\ 0&\frac{3α-1}{α-3}&\frac{-α^2-α-40}{α-3}&\frac{-5α+19}{α-3}&\frac{-9α-49}{α-3}\\ 0&0&\frac{\left(α+4\right)\left(2α-17\right)}{3α-1}&\frac{7\left(α+4\right)}{3α-1}&\frac{3\left(α+4\right)\left(3α-8\right)}{3α-1}\end{pmatrix}$ – Felipe Maia Jun 19 '17 at 19:01
• But I can not think how to discuss the solutions to this result – Felipe Maia Jun 19 '17 at 19:03