When solving an equation, why do you apply the inverse of the given operation to both sides?

So, you are trying to solve for something, and the only way to do that is through applying the inverse of the given operation. This is because you are trying to "undo" everything and get your variable alone.

For example, if you have x+2=4, the only way to get x alone is to subtract 2, but you need to do this to both sides because you want the same equation as when you started. You are just trying to shift everything away from your variable

then the equation will be  x+2-2=4-2 which means x=2
plug it back in to check, 2+2=4, this is correct so you know you have done it right.'

however, if you hadnt done this to both sides watch what happens

x+2-2=4 which means x=4, now check your answer

you would get 4+2=4 which means 6=4 and you know that is not true

Whose mistake? can you give more details to what the question is asking?