Question

When solving an equation why do you apply the inverse on the given operation to both sides

Answer 1

You can do anything you want to one side of an equation.
You'll usually apply an inverse operation to make the side
simpler, for example to get rid of the 'x', or to get rid of the
fractions on that side.

But when you do something to one side of the equation,
you MUST do exactly the same thing to the other side too.
If you don't, then you have changed the equation to a
different one, and the solution to the one you have now
is not the solution to the original one.

Very simple example:

Solve the equation            2x = 2

Correct solution:
Divide each side by  2 :      x = 1

Incorrect solution:
Divide the left side by  2 :    x = 2