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Is |x| + |y| = 0 ?

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by Vincen » Tue Dec 19, 2017 7:20 am
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>
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by ceilidh.erickson » Wed Aug 01, 2018 7:35 am
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 less 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 less 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 absolute value of the other one? This only works if both values are zero! Thus we know that x = 0 and y = 0.

The answer is C.
Last edited by ceilidh.erickson on Thu Aug 02, 2018 5:37 am, edited 1 time in total.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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Is |x| + |y| = 0 ?

by regor60 » Thu Aug 02, 2018 5:31 am
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.
Shouldn't this be negative etc or am i missing something
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edit

by ceilidh.erickson » Thu Aug 02, 2018 5:40 am
regor60 wrote: Shouldn't this be negative etc or am i missing something
You're completely right! Total goof from going too quickly. I've edited accordingly.

The underlying idea hasn't changed, though: if 2 unknowns are each twice the absolute value of the other, they must both be 0.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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