Is |x| + |y| = 0?
Statement 1: x + 2|y| = 0
x = -2|y|.
Case 1: x=0 and y=0.
In this case, |x| + |y| = 0.
Case 2: x=-2 and y=1.
In this case, |x| + |y| ≠0.
INSUFFICIENT.
Statement 2: y + 2|x| = 0
y = -2|x|.
Case 1: x=0 and y=0.
In this case, |x| + |y| = 0.
Case 3: x=1 and y=-2.
In this case, |x| + |y| ≠0.
INSUFFICIENT.
Statements combined:
Squaring x = -2|y|, we get:
x² = 4y².
Squaring y = -2|x|, we get:
y² = 4x².
Substituting y² = 4x² into x² = 4y², we get:
x² = 4(4x²)
x² = 16x²
0 = 15x²
0 = x²
0 = x.
Since x=0, we know that y=0, as indicated in Case 1 above.
Thus, |x| + |y| = 0.
SUFFICIENT.
The correct answer is C.
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