I find that I usually think about this step in the problem as a black box, into which I put a set of equations and out of which comes a set of solutions. This is particularly true if the algebra involved includes many steps.
This is good in a lot of ways and bad in some others, but putting that aside, I'm fascinated that this step seems to come up in all areas of math. Is it possible to separate math from algebra? Are there any branches of math where long chains of algebra are not found or uncommon?