-6x-3=3-6x

Question

What do you do when variables are the same number and if you add to one side, it will make other side 0.

Answer 1

Your equation is -6x-3=-6x+3

There is no number which gives the same result when 3 is subtracted from it as when 3 is added to it.

There can be no solution.  What you have written contradicts the definition of addition.