agree with the original solution as the best method though the description is a little long-wided.

if you take 1 - you can simplify to:

x=-3y

now we know that x and y will have opposite signs so by multiplying them we will have a negative number. We don't really care WHICH one is -ve and +ve just that their product will be -ve. Now you can substitute the above into the given equation to get:

|y| + |-3y| = 32
|y|+3|y| = 32
4|y| = 32
|y| = 8

now |x| = |-3y|

so |x| = 3|y| = 3*8 = 24

now we can multiply |x|*|y| and we know the sign is -ve from before

hope that didn't confuse things too much

Well, this is how I did it. Remember: In DS questions, it is very important to know when to stop calculating! You have to foresee.

Question: What is xy?

Lets look at Statement 1.
-4x -12y = 0
=> x = -3y

This equation tells us that x and y have opposite signs. Regardless of which of the two (x and y) is positive/negative, xy will be negative so we can ignore that. The ratio x:y is 3:1. We know (from the question), that the sum of |x| and |y| is 32. Using the ratio and the sum, we can calculate -|x|*|y| which is equal to xy.

The moment you see that you have the ratio x:y, know that you don't need to know the signs and have the sum of |x| and |y|, you should know that Statement 1 is sufficient.

Statement 2: |x| - |y| = 16

|x| + |y| = 32 (given in the question)

The problem with this statement is that you have absolutely no information about the signs. The moment you see that, you should know that this statement is insufficient.

You can figure out that for the above pair of equations to hold, |x| = 24 and |y| = 8. However, either one of them could be negative or positive (i.e. xy could be negative OR positive). Hence, making it possible to calculate |xy| but not xy.

I solved the pair of equations for |x| and |y| manually.

You know that |x| and |y| should add up to 32 and their difference should be 16. So pick numbers accordingly.

|x| |y| |x|-|y|

30 2 30
26 6 20
24 8 16 (BINGO)