Sometimes I think Mathematica might better be called MetaMathematica, not so much a tool for doing mathematics but a tool for making the tools to do mathematics.
Quite often it will be very useful to "combine and repackage" Mathematica procedures into forms that are more natural and convenient for your application. Writing some definitions and functions along these lines can be very useful. You don't have to limit yourself to the hooks and buttons that are manifest in plain Mathematica.
Maybe there's a mathematical reason why simple operations on equations aren't handled as I'd expect. Mathematica is clearly a powerful tool, but the few times I've tried picking it up in the past I wind up getting stymied by the obtuse syntax for certain simple operations.
In this case, I could probably use the power of the tool all at once by simply treating my set of equations as a unit and asking Mathematica to reason about them as a group and solve my problem directly. The problem is that I, and my audience, could probably gain some insight into the problem by working through a few of the intermediate results. By treating my equations lexically, rather than mathematically, I'm forgoing Mathematica's expertise and only allowing it to ensure I don't make transcription errors.