<s>[INST] inhomogen linear equation system problem I need to find all the answers of this linear equation system.
$$\begin{vmatrix}
1 & 2 & 1 & 3 & 0 & =\beta \\
2&3&2&5&1&=0\\
2&1&1&4&1&=0\\
3&3&2&7&1&=1\\
\end{vmatrix} $$
I tried to solve it like that:
Row 4- row 3->
$$\begin{vmatrix}
1 & 2 & 1 & 3 & 0 & =\beta \\
2&3&2&5&1&=0\\
2&1&1&4&1&=0\\
1&2&1&3&0&=1\\
\end{vmatrix} $$
Row 1- row 4 ->
$$\begin{vmatrix}
0 & 0 & 0 & 0 & 0 & =\beta-1 \\
2&3&2&5&1&=0\\
2&1&1&4&1&=0\\
1&2&1&3&0&=1\\
\end{vmatrix} $$
and if I understand correctly, if a row is in the form of the first row here, all the variable are zero and equals something, the linear equation system don't have an answer.
but I checked that LES in online calculator, and apparently it does have an answer.
Am I getting something wrong here?
 [/INST] As I wrote in the comment. From your work we conclude that the linear system might only possess a solution for $\beta = 1$. As it turns out the solutions for this case are of the form
$$\left(\begin{matrix}
1-2\cdot x_4 \\
2-x_4+x_5 \\
-4+x_4-2\cdot x_5 \\
x_4 \\
x_5
\end{matrix}\right)$$
for $x_4, x_5 \in \mathbb{R}$.
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