Solution to example problem: 3:38

You only have to row reduce the augmented matrix to ROW ECHELON FORM to determine the number of solutions using the methods described.

In the example problem, the matrices were already in at least Row Echelon Form. Normally, you would have to do the row reduction yourself first.

At 2:09 what I was trying to say is that the system could have no solution regardless of whether there are free variables, but once you know the system is consistent (no pivot in augmented column), then figuring out if there are free variables tells you if there is a unique solution or infinitely many.