The simplest answer is they exist to provide a round about way to get to a legitimate real solution.

In the same way negative numbers exist in scenerios that should be impossible.

For example i have 5 apples and on a day I lose 7 I gain 3 5-7+3

Now its impossible to lose 7 apples if you only had 5. And yes you can re-arrange the order. Or you can just go into negative and get a legitimate solution.

The math didn't care that it might have been lost 2 found 2 lose 3 find 1 lose 2.