Here is a solution to an equation however I would have solved it differently and incorrectly for that matter.

Correct:
–x – 1 = 2x – 1
0 = 3x
0 = x

Why does my method not yield the same result:
–x – 1 = 2x – 1
–x = 2x – 1 + 1
–x = 2x

From here I would think that both sides should be divided by x or by -1 but that is obviously incorrect.