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Eh, the modulus answers work and all, but I find them uninspired. And technically, they are depending on an assumption that -7 and 19 are not really what they appear to be (elements of integers, reals, etc) but instead the representative sets that make up the modulus groups. I don't believe it's valid to make this assumption.
I'm not familiar with alephs, so I won't comment too much, but again, an extra assumption that would apply to any such expression renders the specific problem meaningless, and in my opinion, takes the fun out of it.
Of course, it's always a possibility, nay a probability, that OP is a fag. Let us give him the benefit of doubt. One important point that all have failed to mention is that in all of these above algebraic manipulations, commutativity has been assumed. Indeed, given x-7 = x+19, I would say that OP is most assuredly a fag, as we have left associativity in all groups, but we cannot ever assume commutativity.
My proposal is to find a group over some set or subset of integers/reals/etc that legitimately contain the numbers 7 and 19, and a group operation which we will denote with + that satisfies the given equation, ideally such that it doesn't satisfy all equations (like using alephs, or something like mod 1)
