You seem to have a fundemental misunderstanding of negative numbers which really isn't helping.
In the example you gave:
9x - 4 = 63
It can be written like this
-4 + 9x = 63
They're exactly the same thing. And in either example you would add 4 to both sides of either equation.
The general rule is, if there's a negative number in there (x-4), you more than likely have to add to both sides, if there's a positive (x+4) number in there, you'll have to subtract from both sides.
Really, all these are are basic number puzzles. You keep eliminating as much as you can until you can't go any further, and that's the answer.