Given that I have an augmented matrix in Row Echelon Form or Reduced Row Echelon form, and the bottom row(s) contain only zeroes.
Generally speaking (disregarding fringe cases if any?):
Why do rows of only zero imply the presence of a free variable?
Can you have a free variable without the bottom row(s) containing only zeroes?
Does the presence of a free variable always mean you have an infinite number of solutions?