I have the following vectors: $$v_{1}=(2,1,-1),\,v_{2}=(-m,-1,3),\,v_{3}=(-3,2,m+1),w=(m+1,m,-1)$$
And I'm trying to understand for which $m\in \mathbb{R}$ we get $v_{1}x+v_{2}y+v_{3}z=w$.
In order to determinate it, I tried to solve the following matrix:
$$\begin{pmatrix}2 & -m & -3 & m+1\\ 1 & -1 & 2 & m\\ -1 & 3 & m+1 & m-1 \end{pmatrix}\to \begin{pmatrix}1 & -1 & 2 & m\\ 2 & -m & -3 & m+1\\ -1 & 3 & m+1 & m-1 \end{pmatrix}\to \begin{pmatrix}1 & -1 & 2 & m\\ 0 & -m+2 & -7 & -m+1\\ 0 & 2 & m+3 & 2m-1 \end{pmatrix}$$
then I changed $m=2$ and I got:
$$\begin{pmatrix}1 & -1 & 2 & 2\\ 0 & 0 & -7 & -1\\ 0 & 2 & 5 & 3 \end{pmatrix}\overset{R_{2}\leftrightarrow R_{3}}{\to}\begin{pmatrix}1 & -1 & 2 & 2\\ 0 & 2 & 5 & 3\\ 0 & 0 & -7 & -1 \end{pmatrix}$$
meaning its has a solution. Then I assumed that $m\neq 2$, so we get:
$$\begin{pmatrix}1 & -1 & 2 & m\\ 0 & -m+2 & -7 & -m+1\\ 0 & 2 & m+3 & 2m-1 \end{pmatrix}$$
But now I'm not sure what to do next. how should I continue from here?