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

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If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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New post Updated on: 18 Feb 2013, 06:57
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If x+|x|+y=7 and x+|y|-y=6 what is x+y=?

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

Originally posted by guerrero25 on 18 Feb 2013, 06:54.
Last edited by Bunuel on 18 Feb 2013, 06:57, edited 1 time in total.
Renamed the topic and edited the question.
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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New post 18 Feb 2013, 07:25
22
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guerrero25 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


If x<0 and y<0, then we'll have x-x+y=7 and x-y-y=6. From the first equation y=7, so we can discard this case since y is not less than 0.

If x>=0 and y<0, then we'll have x+x+y=7 and x-y-y=6. Solving gives x=4>0 and y=-1<0 --> x+y=3. Since in PS questions only one answer choice can be correct, then the answer is C (so, we can stop here and not even consider other two cases).

Answer: C.
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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New post 24 Feb 2013, 00:59
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There can be only four different cases:

Case 1: x<0 , y>0
If this is true then |x| = -x & |y|= y
Equation would bcom : x-x+y = 7 & x+y-y=6 , which means y=7 & x= 6 which can not be true as we assumed x<0, thus rejected

Case 2 : x<0 , y<0
If this holds trues then |x| = -x & |y|=- y
Equation would bcom : x-x+y = 7 & x-y-y=6 , which means y=7 & x= 20 which can not be true as we assumed x<0,y<0 thus rejected

CASE 3 : x>0 , y >0
If this holds trues then |x| = x & |y|= y
Equation would bcom : x+x+y = 7 & x+y-y=6 , which means y=-5 & x= 6 which can not be true as we assumed ,y>0 thus rejected

Case 4 : x>0, y <0
If this holds trues then |x| = x & |y|=- y
Equation would bcom : x+x+y = 7 & x-y-y=6 , solving these equations we get y=-1 & x= 4 which can be true as it meets our assumption of x>0 & y<0, thus accepted

therefore answer x+y= 4-1= 3 hence C.
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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New post 18 Feb 2013, 07:22
2
2
guerrero25 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


for x>0 ; y>0 => x=6, y=-5 ------ [1]
for x>0 ; y<0 => x=4, y=-1 => x+y = 3 -------- [2]
for x<0 ; y>0 => x =6, y=7 ----------- [3]
for x<0, y<0 => y=7, x =20 ----------- [4]

Hence, x+y = 3
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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New post 18 Feb 2013, 08:28
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Bunuel wrote:
guerrero25 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


If x<0 and y<0, then we'll have x-x+y=7 and x-y-y=6. From the first equation y=7, so we can discard this case since y is not less than 0.

If x>=0 and y<0, then we'll have x+x+y=7 and x-y-y=6. Solving gives x=4>0 and y=-1<0 --> x+y=3. Since in PS questions only one answer choice can be correct, then the answer is C (so, we can stop here and not even consider other two cases).

Answer: C.

Hi Bunuel
Adding both eqn we get 2x +IxI+IyI = 13
Now considering X <0 and y>0
2x-x +y = 13
we get x+y =13
Hence answer should be E

Correct me if i am wrong!!!

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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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New post 18 Feb 2013, 08:37
2
Archit143 wrote:
Bunuel wrote:
guerrero25 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


If x<0 and y<0, then we'll have x-x+y=7 and x-y-y=6. From the first equation y=7, so we can discard this case since y is not less than 0.

If x>=0 and y<0, then we'll have x+x+y=7 and x-y-y=6. Solving gives x=4>0 and y=-1<0 --> x+y=3. Since in PS questions only one answer choice can be correct, then the answer is C (so, we can stop here and not even consider other two cases).

Answer: C.

Hi Bunuel
Adding both eqn we get 2x +IxI+IyI = 13
Now considering X <0 and y>0
2x-x +y = 13
we get x+y =13
Hence answer should be E

Correct me if i am wrong!!!

Archit


OA is C, not E, so yes your solution is not correct.

If x<0 and y>0, then we'll have x-x+y=7 and x+y-y=6. From the second equation x=6, so we can discard this case as y here is not more than 0.
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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New post 18 Feb 2013, 09:23
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Hi Bunuel
Is there any specific reason for applying x<0 and y>0 separately to the equation. I mean the signs could have been applied to the sum of both equations.
Another question , why did you not apply the both the signs to one equation only...

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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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New post 18 Feb 2013, 20:57
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guerrero25 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


You can also solve this question using some brute force.

Notice x+|y|-y=6
The first thing that comes to mind is that if y is positive or 0, x = 6. But when we put x = 6 in x+|x|+y=7, we get y negative. So y cannot be positive or 0. y must be negative. So, x+|y|-y=6 becomes x - 2y = 6

Since most options are positive values, it is very likely that x is positive so x+|x|+y=7 becomes 2x+y = 7. You can obtain 7 by subtracting 1 from 8.
If x = 4 and y = -1, both equations are satisfied.
So x+y = 4 - 1 = 3

Answer (C)
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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New post 18 Mar 2013, 12:28
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VeritasPrepKarishma wrote:
y must be negative. So, x+|y|-y=6 becomes x - 2y = 6


I don't understand the reasoning behind this statement.

If y is negative, then x+|y|-y=6 becomes x +|-y|-(-y)=6 therefore x + y + y = 6 finally becomes x +2y=6

The absolute value of a negative integer is a positive value, and the subtraction of a negative integer results in an addition.

What am I missing?
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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New post 18 Mar 2013, 13:24
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Reevak wrote:
VeritasPrepKarishma wrote:
y must be negative. So, x+|y|-y=6 becomes x - 2y = 6


I don't understand the reasoning behind this statement.

If y is negative, then x+|y|-y=6 becomes x +|-y|-(-y)=6 therefore x + y + y = 6 finally becomes x +2y=6

The absolute value of a negative integer is a positive value, and the subtraction of a negative integer results in an addition.

What am I missing?


Reevak, the algebraic definition of the modulus is :

- If x > 0 then |x| = x > 0 ; (1)
- If x < 0 then |x| = -x > 0 ; (2)

The reason why your statement is incorrect is because you altered the equation.

You can't put a "-" sign to a variable to say that it's negative. In fact, in our case, since y is negative, putting a "-" sign to it will make it positive, thus altering your equation and guiding you to a wrong answer.

What you did is the same as doing the following :

y = - 3 is negative. Therefore - y (which is still negative according to your syntax) will be equal to -y = -(-3) = 3 which is NOT negative.

So be careful with your computations.

Now let's get back to your issue :

Since y is negative ( y < 0 ), then according to (2) we'll have |y| = - y.

Injecting it into our equation yields : x+|y|-y = 6 => x - y - y = 6 => x - 2y =6. Hence the result.

You should be fine if you remember the definition of the modulus. I know it can be tricky at first, but once you get used to it, it'll become second-nature to you :)

Hope that helped.
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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New post 18 Mar 2013, 13:44
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The first thing you should keep in mind when facing modulus (absolute value) problems is the definition of the modulus :

- If x > 0 then |x| = x > 0 (1) ;
- If x < 0 then |x| = - x > 0 (2) ;
-|x| IS ALWAYS POSITIVE NO MATTER x ! ;


Now let's solve the problem : (It's a pretty lengthy solution but I'd rather not miss anything :) )

We need to find the value of x+y knowing that : x+|x|+y=7 and x+|y|-y=6

Now considering the definition of the modulus, we have 4 cases :

- Case 1 : x > 0 and y > 0

According to (1) we'll have :

x+|x|+y = 7 => x+x+y = 7 => 2x + y = 7 => 12 + y = 7 => y = -5 (which is contradictory with the fact that y is positive)
x+|y|-y = 6 => x+y-y = 6 => x = 6

- Case 2 : x > 0 and y < 0

According to (1) and (2) we'll have :

x+|x|+y = 7 => x+x+y = 7 => 2x + y = 7 (E1)
x+|y|-y = 6 => x-y-y = 6 => x - 2y = 6 (E2)

Since we have two equations, we can manipulate them to find the value of x and y separately. In this case, multiplying both (E1) and (E2) by 2 then substracting each other will yield : x = 4 and y = - 1 (which is consistent with the fact that x is positive and y is negative). So x+y = 3 which is answer choice C.

But we're not stopping here :)

We have to make sure that the other cases don't contradict what we've found.

- Case 3 : x < 0 and y > 0

According to (1) and (2) we'll have :

x+|x|+y = 7 => x-x+y = 7 => y = 7
x+|y|-y = 6 => x+y-y = 6 => x = 6 (which is contradictory with the fact that x is negative)

- Case 4 : x < 0 and y < 0

According to (2) we'll have :

x+|x|+y = 7 => x-x+y = 7 => y = 7 (which is contradictory with the fact that y is negative)
x+|y|-y = 6 => x-y-y = 6 => x - 2y = 6

So all in all, only case 2 was valid in this case and the correct answer is C.

Note : The method may seem lengthy and daunting but when you know the modulus definition and are comfortable manipulating variables, it should be a breeze.

Hope that helped :D
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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New post 18 Mar 2013, 20:44
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Reevak wrote:
VeritasPrepKarishma wrote:
y must be negative. So, x+|y|-y=6 becomes x - 2y = 6


I don't understand the reasoning behind this statement.

If y is negative, then x+|y|-y=6 becomes x +|-y|-(-y)=6 therefore x + y + y = 6 finally becomes x +2y=6

The absolute value of a negative integer is a positive value, and the subtraction of a negative integer results in an addition.

What am I missing?


Say y = -3, what is the value of |y|?
Substitute -3 in place of y

|y| = |-3| = 3
This is what you mean by mod of a negative number is positive. The number becomes positive when you remove the mod sign.
Notice that when y is negative, |y| is not equal to y. It is equal to -y (which, by the way, is a positive number since y is negative)

We define mod as:
|y| = y if y is positive
|y| = -y if y is negative

On the other hand, if I ask you: what is the value of y? It is -3 only
What is the value of -y? It is -(-3) = 3

x+|y|-y=6
If we know that y is negative, |y| = -y (which becomes positive)
x - y - y = 6 (notice that the -y in the original equation stays the same since it has no mod around it)
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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New post 19 Mar 2013, 09:05
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Virgilius wrote:
Reevak, the algebraic definition of the modulus is :

- If x > 0 then |x| = x > 0 ;
- If x < 0 then |x| = -x < 0 ;

VeritasPrepKarishma wrote:
|y| = |-3| = 3
This is what you mean by mod of a negative number is positive. The number becomes positive when you remove the mod sign.
Notice that when y is negative, |y| is not equal to y. It is equal to -y (which, by the way, is a positive number since y is negative)


Thanks a lot guys! Now there is one thing I still not understand, and another that I do understand.

Now I see that if y = - 3 then |-3| = 3 ≠ y
Instead |-3| = 3 = - y (since the minus sign will turn y positive turning the equality true)
This I understand.

However, there is still one thing I don't get.
If y < 0 then shouldn't this equation x + |y| - y = 6 be equal to x - y + y = 6?

Instead, you mentioned it should be x - y - y = 6 but I don't understand why :(
Why - y and not + y?
Is -(-3) not 3?
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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New post 19 Mar 2013, 09:21
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Reevak wrote:
Virgilius wrote:
Reevak, the algebraic definition of the modulus is :

- If x > 0 then |x| = x > 0 ;
- If x < 0 then |x| = -x < 0 ;

VeritasPrepKarishma wrote:
|y| = |-3| = 3
This is what you mean by mod of a negative number is positive. The number becomes positive when you remove the mod sign.
Notice that when y is negative, |y| is not equal to y. It is equal to -y (which, by the way, is a positive number since y is negative)


Thanks a lot guys! Now there is one thing I still not understand, and another that I do understand.

Now I see that if y = - 3 then |-3| = 3 ≠ y
Instead |-3| = 3 = - y (since the minus sign will turn y positive turning the equality true)
This I understand.

However, there is still one thing I don't get.
If y < 0 then shouldn't this equation x + |y| - y = 6 be equal to x - y + y = 6?

Instead, you mentioned it should be x - y - y = 6 but I don't understand why :(
Why - y and not + y?
Is -(-3) not 3?


You don't have to change the "-" sign to a "+" sign, since that sign is proper to the equation and not the variable.

Let's assume that y = - 3 (y is negative) and x = 0

If what you said was true, meaning that if y is negative, x + |y| - y = 6 will be equal to x - y + y = 6, we'll have for this example : 0 - (-3) + (-3) = 0 + 3 - 3 = 0 ≠ 6, which is obviously false.

My point is : the modulus is the only fonction that can change the sign of a variable or an expression based on its related sign (whether the variable or expression is positive/negative). Don't change the signs that are proper to the equation itself !!!

Hope that was clear enough. :)
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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New post 19 Mar 2013, 10:28
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Virgilius wrote:
If what you said was true, meaning that if y is negative, x + |y| - y = 6 will be equal to x - y + y = 6, we'll have for this example : 0 - (-3) + (-3) = 0 + 3 - 3 = 0 ≠ 6, which is obviously false.

My point is : the modulus is the only fonction that can change the sign of a variable or an expression based on its related sign (whether the variable or expression is positive/negative). Don't change the signs that are proper to the equation itself !!!

Hope that was clear enough. :)


Yes, now it's clear!
I was changing the sign of the equation and by doing that I was changing the sign of the variable.
Now I see my mistake.

Thanks Virgilius, you were of great help.

Kudos :)
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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New post 19 Mar 2013, 10:31
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Reevak wrote:
Virgilius wrote:
If what you said was true, meaning that if y is negative, x + |y| - y = 6 will be equal to x - y + y = 6, we'll have for this example : 0 - (-3) + (-3) = 0 + 3 - 3 = 0 ≠ 6, which is obviously false.

My point is : the modulus is the only fonction that can change the sign of a variable or an expression based on its related sign (whether the variable or expression is positive/negative). Don't change the signs that are proper to the equation itself !!!

Hope that was clear enough. :)


Yes, now it's clear!
I was changing the sign of the equation and by doing that I was changing the sign of the variable.
Now I see my mistake.

Thanks Virgilius, you were of great help.

Kudos :)


No worries :)

It's important for anyone prepping the GMAT to avoid these little confusions. I was a little apprehensive that my explanation wasn't clear enough for you. But I'm glad it was in the end.

Good luck.
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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New post 25 Apr 2013, 23:53
1
Bunuel wrote:
guerrero25 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


If x<0 and y<0, then we'll have x-x+y=7 and x-y-y=6. From the first equation y=7, so we can discard this case since y is not less than 0.

If x>=0 and y<0, then we'll have x+x+y=7 and x-y-y=6. Solving gives x=4>0 and y=-1<0 --> x+y=3. Since in PS questions only one answer choice can be correct, then the answer is C (so, we can stop here and not even consider other two cases).

Answer: C.



Hi Bunuel,
I didn't understand how the second equation became x-2y=6. If y<0, won't it be x+l-yl-(-y)=6 => x+2y=6 ?
Please explain.
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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New post 26 Apr 2013, 00:51
1
sharmila79 wrote:
Bunuel wrote:
guerrero25 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


If x<0 and y<0, then we'll have x-x+y=7 and x-y-y=6. From the first equation y=7, so we can discard this case since y is not less than 0.

If x>=0 and y<0, then we'll have x+x+y=7 and x-y-y=6. Solving gives x=4>0 and y=-1<0 --> x+y=3. Since in PS questions only one answer choice can be correct, then the answer is C (so, we can stop here and not even consider other two cases).

Answer: C.



Hi Bunuel,
I didn't understand how the second equation became x-2y=6. If y<0, won't it be x+l-yl-(-y)=6 => x+2y=6 ?
Please explain.


No! When \(y<0\), then \(|y|=-y\), thus \(x+|y|-y=6\) becomes \(x-y-y=6\) --> \(x-2y=6\).
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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New post 14 Jun 2013, 08:24
Ok, so here is my question:

Let's say we look for all the cases of x,y being positive and negative (four in total)

(x is negative)

x+|x|+y=7 x+|y|-y=6
x+-x+y=7 x+y-y=6
y=7 x=6


If x is negative, why don't we plug in -x for all values of x in both equations (i.e. x+|x|+y=7 ==> -x+-x+y=7)
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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New post 14 Jun 2013, 08:30
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WholeLottaLove wrote:
Ok, so here is my question:

Let's say we look for all the cases of x,y being positive and negative (four in total)

(x is negative)

x+|x|+y=7 x+|y|-y=6
x+-x+y=7 x+y-y=6
y=7 x=6


If x is negative, why don't we plug in -x for all values of x in both equations (i.e. x+|x|+y=7 ==> -x+-x+y=7)


That is not a valid solution:

in your case you are considering x<0 and y>0 so
x+|x|+y=7 x+|y|-y=6
x+-x+y=7 x+y-y=6
\(y=7\) \(x=6\)

but \(x=6\) is not a valid option because you are considering negative values for x.

If x is negative ONLY |x|=-x you cannot change the sign of all Xs
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?   [#permalink] 14 Jun 2013, 08:30

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

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