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If x+|x|+y=7 and x+|y|-y=6 what is x+y=?......Tough question

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by Jay@ManhattanReview » Mon Oct 02, 2017 12:35 am
Mo2men wrote:If x+|x|+y=7 and x+|y|-y=6 what is x+y=?

A. 1
B. -1
C. 3
D. 5
E. 13

OA: C
Source: Magoosh
This has to do with whether x and y are positive, negative, or 0.

Case 1: Say x > 0 and y > 0
x+|x|+y=7 => 2x + y = 7 and x+|y|-y=6 => x = 6. Thus, from 2x + y = 7, we get y = -5 (Not positive!). Thus, y cannot be positive.

Case 2: Say x > 0 and y < 0
x+|x|+y=7 => 2x + y = 7 and x+|y|-y=6 => x - y -y = 6 => x - 2y = 6. Thus, from 2x + y = 7 and x - 2y = 6, we get y = -1 and x = 4 => x + y = 3.

The correct option is C.

We need not take it further as this is a PS question, thus, only one answer must be correct.

Hope this helps!

-Jay

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by [email protected] » Mon Oct 02, 2017 7:51 pm
Hi Mo2men,

There are some subtle Number Property rules in this question (which you'll have to do a little bit of work to define), but you can ultimately use a bit of 'brute force' math to get to the correct answer.

We're told that:
X + |X| + Y = 7
X + |Y| - Y = 6

Notice the role that the absolute value plays in each equation:
-In the first equation, IF X is negative or 0, then Y = 7.... but plugging Y=7 into the second equation gives us X = 6 (and THAT does not match X being negative or 0). Thus, X CANNOT be negative or 0.

-In the second equation, IF Y is positive or 0, then X = 6.... but plugging X = 6 into the first equations gives us Y = -5 (and THAT does not match Y being positive or 0). Thus Y CANNOT be positive or 0.

Now we know that X MUST be positive and Y MUST be negative. Since the answer choices are all simple integers, let's see what happens when we check simple integer values for X and Y (starting with the first equation)....
X=1, Y=5 --> doesn't fit the restriction that Y is negative.
X=2, Y=3 --> doesn't fit the restriction that Y is negative.
X=3, Y=1 --> doesn't fit the restriction that Y is negative.
X=4, Y=-1 --> fits BOTH equations, thus X+Y = 3

Final Answer: C

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Rich
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