[Math] Find $x$ and $y$ in a simple matrix

Question

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.

Answer 1

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

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