What are we solving with row reduction? Why are row reductions so useful in linear algebra? It is easy to get lost in the mechanical solving of equations. I know I can get a matrix into reduced row echelon form. But what are the outcomes of this? What can this mean?

i think of gaussian elimination or row reduction of a matrix A as giving you a simpler basis for the row space. it is only a small step from there by looking at the reduction of augmented matrix to find the compatibility condition on the right hand side b and solve ax = b
in that case.
