A is n by m binary matrix
i := 1 // row and column index
for i := 1 to m do // for every column 
// find non-zero element in column i, starting in row i:
  maxi := i 
  for k := i to n do 
    if A[k,i] = 1 then 
      maxi := k 
  end for
  if A[maxi,i] = 1 then 
    swap rows i and maxi in A and b, but do not change the value of i 
    //Now A[i,i] will contain the old value of A[maxi,i], that is 1
    for u := i+1 to m do 
      Add A[u,i] * row i to row u, do this for BOTH, matrix A and RHS vector b
      //Now A[u,i] will be 0
    end for
  else 
    declare error – more than one solution exist
  end if
end for 
if n>m and if you can find zero row in A with nonzero RHS element, then
  declare error – no solution.
end if 
// now, matrix A is in upper triangular form and solution can be found
use back substitution to find vector x