This is perhaps very naive, but after studying linear algebra from Axler and more, it is only in the first chapter of Hoffman and Kunze that I am met with finally understanding why it's justified that we do row operations etc. to solve a system of linear (I will forgo "linear" in what follows) equations: because equivalent (in the sense of being linear combinations of one another) systems of equations have precisely the same set of solutions (Theorem 1, Hoffman and Kunze). It seems to me that this gives a sufficient condition that two systems of equations have the same set of solutions.
My question is about whether this condition is also necessary. Put differently, is it true that if two systems of equations have the same solutions, then they are equivalent (obtainable as linear combinations of one another)? I see that Hoffman and Kunze ask about the $2\times 2$ case in a problem I am about to work, but is it true in the general $n \times m$ case?