Sorry, this is just a trivial observation, but if you're going to accept

x + x + 100 = 500

and

3x - x = 400

you can probably list any variety along the lines of

(n)x - (n - 2)x + y = 400 + y

so clearly that isn't a finite transformation space.

Now, these transformations might look trivial, but this problem is too. So there's an interesting question of bounding your algebraic transformations to a subset of all possible transformations that still make all possible solutions available, which ultimately doesn't answer yours.