Is |x| + |y| = 0 ?

ceilidh.erickson wrote:

Vincen wrote:Is |x| + |y| = 0 ?

(1) x + 2|y| = 0
(2) y + 2|x| = 0

The OA is C.

How can I use both statements together to get a conclusion here? I don't have it clear. <i class="em em-confused"></i>

ABSOLUTE VALUE is the distance from 0, so it must always be greater than or equal to 0. For the sum of two absolute values |x| + |y| to equal zero, it must be the case that both are equal to zero.

Target question: are both x and y equal to 0?

(1) x + 2|y| = 0
Test values to try to get non-zero values for x and y:
x = -2
y = 1
-2 + 2|1| = 0
Answer to target question: no

Can we get a "yes" answer as well? Certainly - if x and y both equal 0, the statement holds true. If we can get a "no" or a "yes," this is insufficient.

(2) y + 2|x| = 0
Here, the same logic applies as in statement 1, just with the variables reversed, so this must be insufficient as well.

(1) and (2) together
To combine the statements, first rearrange them:
If x + 2|y| = 0, then
x = 2|y|
Thus, x must be greater than or equal to 0 (if equal to 2 times some absolute value), and it's twice the absolute value of y.

If y + 2|x| = 0, then:
y = 2|x|
Thus, y must be greater than or equal to 0 (if equal to 2 times some absolute value), and it's twice the absolute value of x.

How can two values each be equal to twice the other one? This only works if both values are zero! Thus we know that x = 0 and y = 0.

The answer is C.