If I want to solve the system above, I first have to rewrite it like below. However, are there any particular rules that for instance says, that x has to come first, then y and then z, or can you decide the order by yourself and still make it work?
The reason I am asking this question, is because I almost always see that people use the same order, where x comes first, then y and then z.
It might seem like a dumb question, but I don't see anyone talking about this anywhere.
EDIT: If you can freely rewrite the matrix as you want, would there then be a way to rewrite it, that would make it easier to solve the system?
When you write your system of equations as a matrix and then solve, what you are implicitly doing is to consider the matrix form of the system, which is
$$A\cdot\begin{pmatrix}x\\y\\z\end{pmatrix}=b,$$ where $A$ is the matrix of coefficients of the variables and $b$ is the column vector of independent terms.
In the system above, the first column of $A$ contains the coefficients which multiply with $x$, the second column contains the coefficients which multiply with $y$, and the third column goes analogously with $z$. This just happens due to the definition of the matrix product.
You can always change the order of the variables if you wish so, but you must do the same change to the columns of the matrix, so that the final equations, after computing the matrix product, are the same. For example, if you swap $x$ and $y$ then you must swap the first and second columns.
Analogously, you can change the rows of the matrix if you wish, but then you'll have to change the components of the $b$ vector in the same way, in order for the system to keep being the same as before.