<s>[INST] Find $x$ and $y$ in a simple matrix I've just started learning matrices, but I'm not sure how to go about solving this:
$\begin{pmatrix} 1 & 2 & 0 \\ 2 & 0 & 1\\1&0&2\end{pmatrix}$
$\begin{pmatrix} 0\\ x\\ y\end{pmatrix}$ $=$ $\begin{pmatrix} 6\\2\\4\end{pmatrix}$
I'm told to use the gaussian method, but I'm getting confused with the $\begin{pmatrix} 0\\ x\\ y\end{pmatrix}$  column, every example video online that shows how to solve matrices, does not have such a column, I'm not sure what to do, any help would be appreciated. 
 [/INST] Just proceed with Row Reduction as normal.   If it helps, replace the zero with $z$.  If the system is consistent you will conclude that $z=0$.
Can you reduce:
$$\begin{pmatrix} 1 & 2 & 0 \\ 2 & 0 & 1\\1&0&2\end{pmatrix}\begin{pmatrix} z\\ x\\ y\end{pmatrix} = \begin{pmatrix} 6\\2\\4\end{pmatrix}$$
To

 $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\0 & 0 & 1\end{pmatrix}\begin{pmatrix} z\\ x\\ y\end{pmatrix} = \begin{pmatrix} 0\\3\\2\end{pmatrix}$$

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