L. Independence and existence of solution

Suppose I have the following system of equations:

$\displaystyle Ax=b$

Where A is an M by N matrix; M>N.

Now suppose that the columns of A posses linear independence. What restrictions should we apply to b to guarantee the existence of a solution x to system. How can we compute this solution?

I believe we have to apply reduction to leave the matrix in upper triangular form, but no idea how to do it.

Help!