Dividing Equations

There is no set way to attack a given problem. The GMAT is a timed test and ultimately, all the test-takers will be able to figure out a problem given infinite time. You're trying to train your mind to identify the tricks and substitutions that would eventually help you to solve a problem in time, if not actually gain time to use on another problem from something like this. Practice is the only way of really getting through. If you're able to solve something your way in under 2 minutes, it's fine. But keep an alternate idea in mind because it might come in handy in another situation where your method is infeasible for some reason.

That's exactly the reason I posted, I was able to solve it in about a minute. I just thought the method the book gave me may be something I could learn in case my method did not work.

In general, honing your pattern recognition skills to seek out "chunks" of information can be quite useful, as it often saves time and brain space Here, the (a+b) chunk is common to both, and if you saw that you could combine the equations that way to eliminate variables (the book solution is the multiplication version of the stack & add/subtract method for systems of equation), awesome. If not-- no worries--substitution will still work, and it's a waste of time to pull your hair out if you don't ID a "cool" way to do a problem quickly.

Whenever you have the option, though, some elegant stacking-and-eliminating (or chunking to allow you to eliminate) will cut out a few steps; substitution tends to take longer.

To zoom out to the bigger picture, that ability to ID chunks/patterns in unfamiliar info is what will speed you along on the GMAT, and something that all of the highest scorers do. (Splitting/resplitting on Sentence Correction questions is an example of this. And at the heart of it, that's what all the test prep companies are trying to do-- prepare you to recognize patterns in new material-- GMAT questions on your test day-- by analyzing patterns in old material--retired GMAT questions).

Bigred, you already seem to be aware of this; bravo to you for recognizing that a solution might contain a future-applicable tool rather than being content with a simple explanation. Good luck with your prep!