Gaussian Elimination has contradicting results when done in different ways

Question

I am having trouble with the Gaussian Elimination algorithm, doing it in 2 different ways, but still doing it properly leads to contradicting results, one stating that we have 1 solution and the other one telling us that we have infinite solutions

I know this should not be possible, but I cannot explain myself which of those two is wrong, and more importantly why one of them is wrong

first solution (note: first I am doing $\text{IV} - \text{II}$, then I am doing $\text{II} - \text I$) then second solution:

Answer 1

It is somewhat hard to understand what you're doing when all that you're showing is a picture of some numbers with no explanation of what you think you're doing with those numbers.

Instead of posting images, you really ought to show your calculations directly in the post. The fancy MathJax formatting we usually use here takes some time to learn, but what you can do instead is to indent each line with four spaces; then you can do ASCII art instead, which would be quite sufficient for your purpose. (I'll do that here as a demonstration).

The first of your pictures contains something like

 1_  1   1   1 | 7
 0 |_0_  1   3 | 8
 0   0 |_1_  5 | 10
 0   0   0 | 2 | 2

where the zig-zag line seems to suggest that you think it's a row echelon form. But it isn't -- the $0$ in the second diagonal element cannot be a pivot element (pivots are supposed to be $1$ or at least nonzero), so the pivot element of the second row is in the third column. Therefore all of the third column below it should be cleared, with the next pivot (if any) only appearing in the fourth column. That would give you

 1___1_  1   1 | 7
 0   0 |_1_  3 | 8
 0   0   0 |_1_| 1
 0   0   0   0 | 0

as the row echelon form.