we have a system of linear equations as such:
$$ x+2y+(a-1)z=1\\-x-y+z=0\\-ax-(a+3)y-az=-3\\-ax-(a+2)y +0\cdot z=a^2 -5a-2$$

and i have to find the solution in $\mathbb{R}$ and $\mathbb{Z\_{5}}$ so i have no problem for $\mathbb{R}$ i get the matrix
$$ \left(\begin{matrix}
1 & 2 & a-1 & 1 \\
0 & 1 & a & 1 \\
0 & 0 & a & 0 \\
0 & 0 & a & a^2-5\cdot a
\end{matrix}\right)$$ but the questions i have are as follows:

1. can i use what i found for the augmented matrix and the discussion by parameter a in $\mathbb{R}$ to deduce $\mathbb{Z\_5}$?

2.or is there some other way i must reduce to row echelon form for $Z\_5$ and then have the discussion for parameter a?

3. If i had an 3x3 or 4x4 system to solve over a low prime $\mathbb{Z\_{p\_{1}}}$ and $\mathbb{Z\_{p\_{2}}}$ (eg 5 and 7) how would i go about doing it with the matrix gauss elimination?could i use the same augmented matrix and reduce it to row echelon over $\mathbb{R}$ and then use that augmented matrix for the rest like above or not?

4.if i recall correctly there was a theorem about the rank of the original matrix and augmented that says something about the number of solutions but i do not recall how that would help me find solutions just eliminate the a's where there is none?