Data Sufficiency

CaptainM wrote:I could figure out that Option A and B individually are insufficient .On combining (1) and (2) I got an equation
21.5x + 10.25y=129

Now How do I save time to infer that I will get only 1 pair of value for x and y.

well, you've got that equation incorrect -- it should be 21.5x + 10.25x. there is no reason to introduce the second variable here, since both values must be the same -- the question prompt makes it clear that the number of full-price shirts is the same as the number of half-price shirts ("For each ... he also purchased a ...").

so, in this particular problem, you've got one linear equation in one variable, so it should be pretty clear that you are going to get a solution.

--

however, yes, there will be cases in which you'll have such equations with two variables.
(here's one such question: https://www.beatthegmat.com/gmat-prep-qu ... t2465.html)

on such equations, the answer to "how can i shortcut the process?" is ... "you can't".
if you have a linear equation to which the solution must be in whole numbers, the ONLY way to be sure that you are correct about its sufficiency/insufficiency is to TEST NUMBERS.

you shouldn't worry about trying to come up with any sort of clever shortcut method to avoid such testing. if you're not convinced, consider the following two equations, in terms of whole number solutions:
5x + 7y = 48
5x + 7y = 47

the first of these has only one whole-number solution (x = 4, y = 4). the second, however, has two of them (x = 1, y = 6, and x = 8, y = 1).

in any case, it's not as though the number testing really takes a lot of time -- if it does, then this probably just means that Excel has killed your ability to do arithmetic by hand, and so you should probably practice arithmetic until you can do it more efficiently.
the gmat is not going to give you one of these problems with obnoxiously large numbers, so you won't have to worry about situations in which number testing is unreasonable in the given timeframe.