If x and y are non-zero integers ...

[eshieh06]

Would appreciate any help you can offer. I'm taking the GMAT on three days from now ...

The question is:

If x and y are non-zero integers and |x| + |y| = 32, what is xy?

(1) -4x - 12y = 0

(2) |x| - |y| = 16

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The CAT solution for statement 1:

(1) SUFFICIENT: Statement (1) can be rephrased as follows:

-4x - 12y = 0
-4x = 12y
x = -3y

If x and y are non-zero integers, we can deduce that they must have opposite signs: one positive, and the other negative. Therefore, this last equation could be rephrased as

|x| = 3|y|

We don’t know whether x or y is negative, but we do know that they have the opposite signs. Converting both variables to absolute value cancels the negative sign in the expression x = -3y.

We are left with two equations and two unknowns, where the unknowns are |x| and |y|:

|x| + |y| = 32
|x| - 3|y| = 0

Subtracting the second equation from the first yields

4|y| = 32
|y| = 8

Substituting 8 for |y| in the original equation, we can easily determine that |x| = 24. Because we know that one of either x or y is negative and the other positive, xy must be the negative product of |x| and |y|, or -8(24) = -192.

...

The correct answer is A.


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Where I get tripped up on this question:

In statement 1, I don't understand why this part of the solution is true: If x and y are non-zero integers, we can deduce that they must have opposite signs: one positive, and the other negative. Therefore, this last equation could be rephrased as: |x| = 3|y|

How can we make that jump?

Thanks for any help you can offer!

[timothy_straub]

Statement II: ????
|x|-|y|=16
24-8 = 16 =(Question) |x| + |y| =32 = |24|+|8| = 32
xy = ?
24(8)=192

Wouldn't Stmt II also be Sufficient??
There are multiple numbers subtracted that = 16 but only one set that add to 32?
It seems that the numbers 24,8 make (II) correct and also make the question correct so, xy= 192. I would think Ans.(D)??

Why is Stmt II Insufficient?Because multiple numbers meet (II) requirements? Not sure how to think about this type of question.
Usually if you can find one instance of disproving the statements, it yields Insufficient but this question states " x and y are non-zero integers" with no constraint?

Any simple examples dealing with tricky DSQ's?

[esledge]

Where I get tripped up on this question:

In statement 1, I don't understand why this part of the solution is true: If x and y are non-zero integers, we can deduce that they must have opposite signs: one positive, and the other negative. Therefore, this last equation could be rephrased as: |x| = 3|y|

How can we make that jump?

I personally wouldn't make that jump, either. It's more intuitive to me to work in the realm of non-absolute values, so I'm always trying to break the variables out of the absolute value bars, not put them back in!

But, let's see, once we get to x = -3y, we have two cases:

If y is negative, x = -3 * neg = pos
If y is positive, x = -3 * pos = neg

In either case, x and y have opposite signs. Therefore, xy is negative. At that point in the solution, it seems they rephrased to |x| = 3|y| to ignore the sign difference and focus on the ratio and determine the relative values of x and y. Don't fret if that doesn't match the way you think.

I see the sign difference as very relevant to this problem, as the original constraint (|x|+|y|=32) has solution sets with not only uncertain values, but also uncertain signs: (x,y) = (+/-31,+/-1),....,(+/-25,+/-7), (+/-24, +/-8),....,(+/-1,+/-31).

(1) tells us not only that the (x,y) solution pair is (+/-24,+/-8), but specifically that it is either (-24,8) or (24,-8). The xy value must be -(8)(24) either way.

Statement II: ????
|x|-|y|=16
24-8 = 16 =(Question) |x| + |y| =32 = |24|+|8| = 32
xy = ?
24(8)=192

Wouldn't Stmt II also be Sufficient??
There are multiple numbers subtracted that = 16 but only one set that add to 32?
It seems that the numbers 24,8 make (II) correct and also make the question correct so, xy= 192. I would think Ans.(D)??

Why is Stmt II Insufficient?Because multiple numbers meet (II) requirements? Not sure how to think about this type of question.
Usually if you can find one instance of disproving the statements, it yields Insufficient but this question states " x and y are non-zero integers" with no constraint?

Any simple examples dealing with tricky DSQ's?

You are missing the other solutions to both constraints. These have the same absolute values for x and y respectively, but various sign combinations:
x = 24, y = 8: |24| + |8| = 32 and |24| - |8| = 16
x = -24, y = 8: |-24| + |8| = 32 and |-24| - |8|= 16
x = 24, y = -8: |24| +|-8| = 32 and |24| - |-8| = 16
x = -24, y = -8: |-24| + |-8| = 32 and |-24| - |-8| = 16

The product xy can be either 192 or -192-->those two solutions are the reason (2) is insufficient.