If x and y are non-zero integers and |x| + |y| = 32, what is xy?
(1) -4x - 12y = 0 (2) |x| - |y| = 16
Statement 1: -4x - 12y = 0.
-4x = 12y
x = -3y.
Substituting x= -3y into |x| + |y| = 32, we get:
|-3y| + |y| = 32
3|y| + |y| = 32
4|y| = 32
|y| = 8
y = 8 or y = -8.
If y=8, then x = -3*8 = -24, and xy = (-24)(8) = -192.
If y= -8, then x = -3*(-8) = 24, and xy = -8*24 = -192.
Since xy = -192 in each case, sufficient.
Statement 2: |x| - |y| = 16.
Adding this equation to |x| + |y| = 32, we get:
(|x| - |y|) + (|x| + |y|) = 16 + 32
2|x| = 48.
|x| = 24
x=24 or x = -24.
This means:
24 + |y| = 32
|y| = 8.
y = 8 or y = -8.
If x=24 and y=8, then xy = 192.
If x= -24 and y=8, then xy = -192.
Since xy can be different values, insufficient.
The correct answer is
A.
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