https://en.wikipedia.org/wiki/Reduction_(complexity)
In practice, there are many cases where people use SAT solvers for other problems. For some examples:
John Langford of Microsoft Research (and formerly of Yahoo! Research) has some some extensive theoretical and practical work here. In particular as they apply to machine learning problems. Check out his page on this[1]. I'd also recommend checking out Vowpal Wabbit[2] which is essentially a working implementation of many of these reductions in ML.
1. https://hunch.net/~jl/projects/reductions/reductions.html
the last one (what's computable) forms the basis for a lot of (all?) cryptography and is filled with all sorts of really fun and not at all hard to master proofs that are just reductions at their core.
statisticians, electrical engineers, data scientists and physicists will all often use the same math (linear systems) but with very different names attached.