I understand why two of the three row operations do not change the solution set of a system: 1. Interchange two rows. (Doesn't make much difference in what order one decides to write down the linear equations does it?) 2. Multiply a row by a scalar. (This step doesn't change the solution set because e.g. writing 2x = 6 instead of x = 3 doesn't change the geometric situation at all) It's the third one that's giving me trouble: 3. Replace one row with the sum of it and a multiple of another row. Even though this is the key method of solving simultaneous equations, I cannot justify to myself that it doesn't change the solution set of a system. Can anyone shed light on this?