Wouldn't know where to start, but I'd have a go at seeing what we've got by attacking it like this.
I would eliminate one of the variables and so get an equation in 2 variables. This will be linear. I would then experiment with plotting it, or something, and see whether there is a constraint that forces a certain solution upon me. Otherwise, you would see there is no unique solution. Unless there is a diophantine equation lurking at the bottom here, in which case I've got this page here:
Solution of Linear Diophantine Equation - ProofWiki
but I don't know whether this will help or if it's way off beam.