Technically, you can choose any variable to be a parameter as long as its value is not completely determined. I tend to go from the right-hand-side of the matrix towards the left-hand-side, as, it appears, you do also. Let's take the following ref augmented matrix as an example:
Method one for back substitution:
Let Then we have the equation or We also have the equation or The solution is then
Method two for back substitution:
Let Then we have the equation or or We also have the equation or , or The solution is then
Question: are these the same solution?
Well, both solutions are the equation of a straight line. So, are they both describing the same straight line? Well, if they're parallel and have at least one point in common, then they are the same line. That the lines are parallel you can tell because the direction vectors (the vectors multiplying the parameters or ) are multiples of each other. I can multiply the direction vector by to get the direction vector. Do they have a point in common? Well, clearly, the point is on the line, which point I got simply by setting . Is that point also on the line? We set
The second component tells us that and you can tell by inspection that that value for also works for the first and third components. Hence, is on both lines, and the lines are the same.
Therefore, the solutions are the same.
What it amounts to is a re-parametrization of the solution. The actual vectors in the solution space are not going to change one way or the other.
Does all that make sense?