Why you are allowed to do any of this
Equals added to equals stay equal. If a triple (x, y, z) makes equation 1 true and equation 3 true, it also makes any multiple of one added to the other true — you added the same amount to both sides. And every such move can be undone by adding the opposite multiple back. Nothing is gained, nothing is lost: the new system has exactly the same solution set as the old one. That is the whole license for elimination.
The classic slip. Both eliminations must remove the same variable, and between them all three equations must get used. Pair 1 & 2, then pair 1 & 2 again, and you have written the same fact down twice — the third equation never got to speak, and you will never pin the point down.