Common GMAT trap: 31X+25Y=280

First of all, 31*x+25*y=280 has infinitely many solutions, not just one.

It will have only one solution x=y=5 if x and y are restricted to positive integers only. In this case 31*x+25*y=280 is a special kind of equation (Diophantine equation) and you can encounter them in problems where x and y must be positive (nonnegative) integers only, for example when they represent # of people/items. Now, trial and error along with some common sense is pretty much the only way you should approach such kind of problems on the GMAT. You won't get some very tough numbers to manipulate with or there will be some shortcut available, based on multiples concept or on the answer choices. So generally you would have to try just couple of values to get the answer.

For example for this problem since 25*y and 280 are both multiples of 5, then 31y, or simply y, is also a multiple of 5. So you need to check ONLY one value: y=5 (since if y=10 then 31y=310>280) and see whether it yields integer solution for x (well if its given that x and y are positive integers and 31*x+25*y=280 then since only possible value of y is 5 then it must yield integer solution for x, in order the given statements to be true).

For more on this type of questions check:
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Hope it helps.