Say we have e.g. the system of linear equations:
$$\begin{bmatrix} 1&0&1\\0&-1&0\\1&0&1\end{bmatrix} \begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}1\\2\\3\end{bmatrix}$$
Why does Augmented Matrix let us ditch \begin{bmatrix}x\\y\\z\end{bmatrix} in order to get:
$$ \left[
\begin{array}{ccc|c}
1&0&1&1\\
0&-1&0&2\\
1&0&1&3
\end{array}
\right] $$
Does this hold for non-linear systems of equations as well?
You don't need to write the name of the variable any more as in the matrix each row represents the coefficient of the variables. So if you can reduce a row to only one single 1 and all other entries of the row to 0. then basically you get the value of the corresponding variable on the right hand side.
For example: as you got the 2nd entry in the 2md row to be -1 you can directly say that the value of y is -2.