I'm finding it hard to read what you've written, in particular I can't make out the word after "when" on the 2nd line.
I'd say the "key" fact here is that the rank of the augmented matrix is less than or equal to the number of equations. I'm not convinced you've justified this sufficiently, but it may be fine.
You also have a logic error: you say "for the system to have an infinite number of solutions r < n", but what you actually need is "if r < n, the system has an infinite number of solutions".
In terms of layout / wording, I think your solution would be a lot clearer if you started off by actively defining m and n:
"Let m be the number of equations in the system, and let n be the number of variables. Then ..."