For the system of equations above, if I were to use matrices to solve for each of the variables, how is the variables matrix ordered? If all the variables in the equations are aligned vertically (as they already are), if I look at one of the equations, do the variables from the left to right correspond to the variables matrix's up and down? In other words, would the variables matrix then be ordered as such: the top variable is z, the middle variable is x, and the bottom variable is y? What is the general rule for orienting the variable matrix, because I can't seem to find one online?
Usually we choose the alphabetical order $x,y,z$ but of course we can also use any others order and in particular $z,x,y$.
In the latter case the matrix form will be
$$\begin{bmatrix}2&1&-1\\-1&2&1\\-3&-1&2\end{bmatrix}\begin{bmatrix}z\\x\\y\end{bmatrix}=\begin{bmatrix}-3\\0\\7\end{bmatrix}$$
which is completely equivalent to
$$\begin{bmatrix}1&-1&2\\2&1&-1\\-1&2&-3\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}-3\\0\\7\end{bmatrix}$$