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

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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] New post 14 Jun 2013, 07:48
Got it! I was getting confused with plugging in negatives throughout but it makes a lot more sense now. Thanks!

Zarrolou wrote:
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] New post 14 Jun 2013, 07:59
Expert's post
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)


Think about this:

Is 2x + 7 = 0 same as -2x + 7 = 0?

You know that these two are different equations and yield different values of x. x can be negative/positive.

On the other hand, how do you solve something like this: 2|x| + 7 = 0
You need the value of x, not of |x|. How will you get the value of x?

You will use the definition of |x|

|x| = x if x >= 0
|x| = -x if x < 0

So you take 2 cases:

Case 1:
x >= 0 so |x| = x
2|x| + 7 = 0 => 2x + 7 = 0
x = -7/2 (this doesn't work since x must not be negative)

Case 2:
x < 0 so |x| = -x
2|x| + 7 = 0 => 2(-x) + 7 = 0
x = 7/2 (this doesn't work either since x must be negative)

So there is no value of x that satisfies 2|x| + 7 = 0
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] New post 01 Jul 2013, 12:20
If x+|x|+y=7 and x+|y|-y=6 what is x+y=?

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

There are four possible cases here I. (x<0, y<0) II. (x<0, y>0) III. (x>0, y<0) IV. (x>0, y>0)

I.) (x<0, y<0)
x+|x|+y=7 x+(-x)+y=7 y=7
x+|y|-y=6 x+(-y)-y=6 x-2y=6
x-2(7)=6 x-14=6 x=20

II.) (x<0, y>0)
x+|x|+y=7 x+(-x)+y=7 y=7
x+|y|-y=6 x+y-y=6 x=6

III.) (x>0, y<0)
x+|x|+y=7 x+(x)+y=7 2x+y=7
x+|y|-y=6 x+(-y)-y=6 x-2y=6
x=6+2y
2(6+2y)+y=7
12+4y+y=7
12+5y=7
5y=-5
y=-1

2x+y=7
2x+(-1)=7
2x=8
x=4

IV.) (x>0, y>0)
x+|x|+y=7 x+x+y=7 2x+y=7
x+|y|-y=6 x+y-y=6 x=6
2(6)+y=7 y=-5

If you notice, III.) is the only one in which the x and y values fall within the tested ranges. (x>0, x=4 y<0, y=-1)

(C)
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] New post 02 Jul 2013, 10:01
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


after some time, working hard realized how easy is this :(
here i go finally

lxl is both -/+ so try plugging them all and remember when u plug lyl , when y is < 0 then lyl is positive and -y is also positive
u r done :)

logic and basic => Gmat magic :)
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] New post 28 Dec 2013, 03:16
My god, so long! Ok i was not 100% into it, but nevertheless, took me 3:40 to solve it!

Really long and hard question!

I think the best explanations were already given!

Answer is C!
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] New post 04 Feb 2014, 22:53
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.


bunuel

When x less than 0 and y less than 0, I get x-x+y=7 and x+y+y =6. Coz if y=-2 then x+mod(-2)-(-2)=6 which gives x+2+2. Pls correct me if wrong. Thanks in advance.

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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] New post 05 Feb 2014, 00:07
Expert's post
nishanthadithya 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.


bunuel

When x less than 0 and y less than 0, I get x-x+y=7 and x+y+y =6. Coz if y=-2 then x+mod(-2)-(-2)=6 which gives x+2+2. Pls correct me if wrong. Thanks in advance.

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When y<0, |y|=-y;
When y>0, |y|=y.

Hence, when y<0, then |y|=-y, thus x+|y|-y=6 becomes x-y-y=6 --> x-2y=6.

For example, if y=-2, then x+|y|-y=6 becomes x+2-(-2)=6 --> x+4=6 (x-2y=6).

Theory on Abolute Values: math-absolute-value-modulus-86462.html

DS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=37
PS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=58

Hard set on Abolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html


Hope it helps.
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If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] New post 30 Mar 2014, 07:32
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


After struggling to understand all the explanations, I solved it my way.

x+|x|+y=7 ------ (1)
x+|y|-y=6 ------ (2)

Take (1),
Case |x| = x,
(1) => x + x + y = 7
or, 2x + y = 7 ----- (A)

Case |x| = -x,
(1) => x - x + y = 7
or, y = 7 ----- (B)

Take (2),
Case |y| = y,
(2) => x + y - y = 6
or, x = 6 ----- (C)

Case |y| = -y,
(2) => x - y - y = 6
or, x - 2y = 6 ----- (D)

Now lets look at (A), (B), (C) and (D).
We can simply ignore (B) and (C) because they can't be solutions as they were not obtained by solving the system of given equations. They are independent of the system. Even if we tried to use them they wouldn't fit the constraints given by equations.
Simple check, try using y = 7 in (2)'s case when y is positive. There is no y in the equation to substitute.

Solving the linear system of equations (A) and (D),
x = 4, y = -1,

Thus, x + y = 3

Last edited by gkashyap on 29 Jun 2014, 12:59, edited 1 time in total.
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] New post 29 Jun 2014, 06:55
May be its a way to solve just check once!

1. x + |x| + y = 7

2. x + |y| + y = 6

now ,

we write it as |x|= 7 - x - y and |y| = 6 + y - x (rearranging the equation)

now if |x|= a then we write it as x = a or x = -a

Similarly,

x = 7 - x - y (2x = 7 - y) and x = x + y - 7 (y = 7) for equation 1

y = 6 + y - x (x = 6 ) and y = x - y- 6 (2y = x - 6) for equation 2


now solving both 2x = 7 - y and 2y = x - 6 w get x = 4 and y = -1 :):)


So x+y = 3 ANS : C
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] New post 29 Jun 2014, 20:23
Expert's post
digishajain wrote:
May be its a way to solve just check once!

1. x + |x| + y = 7

2. x + |y| + y = 6

now ,

we write it as |x|= 7 - x - y and |y| = 6 + y - x (rearranging the equation)

now if |x|= a then we write it as x = a or x = -a

Similarly,

x = 7 - x - y (2x = 7 - y) and x = x + y - 7 (y = 7) for equation 1

y = 6 + y - x (x = 6 ) and y = x - y- 6 (2y = x - 6) for equation 2


now solving both 2x = 7 - y and 2y = x - 6 w get x = 4 and y = -1 :):)


So x+y = 3 ANS : C


There is a flaw here:

From equation 1, you get 2x = 7 - y or y = 7
From equation 2, you get x = 6 or 2y = x - 6

So then we already have 1 value each for x and y, right? y = 7 and x = 6. Then x+y = 13. Why would we solve the the other two equations instead? Also, why can't we solve 2x = 7 - y and x = 6 together to get the values of x and y. How about solving y = 7 and 2y = x-6 together? What made you decide that we must solve 2x = 7 - y and 2y = x - 6 only to get the answer?

When you remove the absolute value sign, remember that it is subject to conditions.

x = 7 - x - y (only if x >= 0) and x = x + y - 7 (only if x < 0) for equation 1

y = 6 + y - x (only if y >= 0) and y = x - y- 6 (only if y < 0) for equation 2

So only if x < 0, y is 7
And only if y >= 0, x is 6.

So x cannot be 6 when y is 7 (since x must be negative if y is 7). And that is the reason x+y is not 13.
This is also the reason the other two equation pairs cannot be solved either.

ALWAYS keep the positive/negative conditions in mind.

Check out tomorrow's post on my blog: http://www.veritasprep.com/blog/categor ... er-wisdom/
It will discuss why it is necessary to keep the conditions in mind.
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] New post 18 Sep 2014, 10:53
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


look at second equation

x+|y|-y=6
if y is positive x = 6
substituting in eq. 1 we get y as negative,therefore y cannot be positive. Hence y has to be negative... (I)

Considering y as negative
eq. 2 becomes x - 2y = 6
this indicates x has to be positive and not equal to 0... (II)
so eq. 1becomes 2x+y = 7

solving both eq's we get x = 4 and y = -1

P.S. : eq = equation
Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?   [#permalink] 18 Sep 2014, 10:53
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