cat question - opposite signs

[Guest]

I am confused as to how they get the parts i've highlighted in bold. Can someone please explain more clearly? Thanks!

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

(1) -4x - 12y = 0

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

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.

(2) INSUFFICIENT: Statement (2) also provides two equations with two unknowns:

|x| + |y| = 32
|x| - |y| = 16
Solving these equations allows us to determine the values of |x| and |y|: |x| = 24 and |y| = 8. However, this gives no information about the sign of x or y. The product xy could either be -192 or 192.

The correct answer is A.

[StaceyKoprince]

FYI The part of the above explanation right below the question applies to statement 1 (looks like the poster didn't copy the entire explanation, just the part of it that pertained to his/her question).

Statement 1: -4x - 12y = 0 If x and y are both positive, then I'd have a negative number (-4x) minus a positive number (12y), which would be an even more negative number. But it equals zero, so they can't both be positive. Try this with real numbers: x = 2, y = 3. -4(2) - 12(3) = -8 - 36 = even more negative.

If both x and y are negative, then I'd have a positive number (-4x) minus a negative number (12y), which would be an even more positive number. But it equals zero, so they can't both be negative. Try this with real numbers: x = -2, y = -3. -4(-2) - 12(-3) = 8 - (-36) = 8 + 36 = even more positive.

The only option I have left is opposite signs.

Taking -4x - 12y = 0, I can simplify that to -x - 3y = 0, or -x = 3y or x = -3y. Since one must be positive and one must be negative, I can also write this with absolute value signs instead: |x| - 3|y| = 0 - that's just something to know, so know that you can do that. That covers the case when x is pos and y is neg AND the case when x is neg and y is pos. Note that the two equations mentioned in the explanation, |x| = 3|y| and |x| - 3|y| = 0, are the same equation - just rearranged.