Solving non homogeneous linear equations

hi, Consider a system of linearly independent, non-homogeneous system of m linear equations in m unknowns having a unique solution. We can solve for the unknowns by Gaussian Elimination in O(n^3); n the number of linear equations in the system; given all the m equations. Is there a solution better than brute force, when there are say m-1 equations and m unknowns? Thanks, Sarad.