Ill-conditioning is not only dependent on the matrix of coefficients. This example [Hamming] shows that it can arise from an interaction between the left and right sides of the system. Let ε be a small real.
3x + 2y + z = 6
2x + 2εy + 2εz = 2 + 4ε
x + 2εy - εz = 1 + ε
(a) Solve the system by hand. Notice that the "'s divide out only because there is an exact cancellation of the integer parts on the right side as well as on the left.
(b) Solve the system by hand, rounding to two decimal places, and with ε = 0.001.