Partial solution for linear system

Hi,

I would to solve overdetermined linear system Ax = b, where A is (mxn) matrix (m > n), b is (mx1) vector, hence x is (mx1) vector. I am not interested in the least square solution,but I am interested in partial solution to the problem. What I mean by partial solution is how many components of x that I can get.

Let say A = [1 0 0; 0 1 1; 0 0 0], b is [ 1 4 4], A is singular but I can obtain the value of first component, which is 1. I would like to know is there an efficient and stable for this problem in matlab.

Currently, my solution is to do rref(A), then check row by row whether it has only one non zero value or not.

Thank you