Transforming a linear matrix equation to Reduced Row Echelon Form (rref) [1] allows solving it very easily or writing a minimal set of variables in terms of others if you don't have enough independent equations or discovering that the set of equations is contradictory. You naturally arrive at this form if you were to do manual Gaussian Elimination, but linear algebra libraries have additional efficient ways to arrive at this depending on the structure of the matrix. So it's common to just ask the algebra system to get it there and then use the matrix either programmatically or by hand depending on what you're trying to do.
[1]: See https://en.wikipedia.org/wiki/Row_echelon_form