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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]

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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]

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14 Jun 2013, 07:59
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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)

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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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Senior Manager Joined: 13 May 2013 Posts: 472 Followers: 3 Kudos [?]: 160 [0], given: 134 Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] ### Show Tags 01 Jul 2013, 12:20 1 This post was BOOKMARKED 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) Manager Joined: 03 Mar 2013 Posts: 91 Location: India Concentration: General Management, Marketing GPA: 3.49 WE: Web Development (Computer Software) Followers: 0 Kudos [?]: 8 [0], given: 6 Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] ### Show Tags 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 Senior Manager Status: Student Joined: 26 Aug 2013 Posts: 265 Location: France Concentration: Finance, General Management Schools: EMLYON FT'16 GMAT 1: 650 Q47 V32 GPA: 3.44 Followers: 2 Kudos [?]: 62 [0], given: 401 Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] ### Show Tags 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! _________________ Think outside the box Intern Joined: 13 Aug 2013 Posts: 3 Followers: 0 Kudos [?]: 0 [0], given: 0 Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] ### Show Tags 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. Posted from my mobile device Math Expert Joined: 02 Sep 2009 Posts: 36567 Followers: 7081 Kudos [?]: 93194 [0], given: 10553 Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] ### Show Tags 05 Feb 2014, 00:07 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. Posted from my mobile device 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. _________________ Intern Joined: 21 May 2013 Posts: 8 Followers: 0 Kudos [?]: 1 [0], given: 7 If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] ### Show Tags 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. Intern Joined: 13 Apr 2014 Posts: 2 Location: United States Concentration: General Management, Sustainability Schools: Wharton '16 GMAT Date: 08-28-2014 GPA: 3.52 WE: Information Technology (Computer Software) Followers: 0 Kudos [?]: 0 [0], given: 2 Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] ### Show Tags 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 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7125 Location: Pune, India Followers: 2134 Kudos [?]: 13653 [0], given: 222 Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] ### Show Tags 29 Jun 2014, 20:23 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. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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

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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
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x+/x/+y=7 and x+/y/-y=6, then x+y=? [#permalink]

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02 Mar 2015, 12:11
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x+/x/+y=7 and x+/y/-y=6, then x+y=?

a. -1
b. 1
c. 3
d. 5
e. 13
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Re: x+/x/+y=7 and x+/y/-y=6, then x+y=? [#permalink]

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02 Mar 2015, 14:02
Hi amianik,

It's not clear what 'symbol' you're using there. Are those supposed to be absolute value signs?

Are the equations supposed to be....
X + |X| + Y = 7 and X + |Y| - Y = 6

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# Special Offer: Save $75 + GMAT Club Tests 60-point improvement guarantee www.empowergmat.com/ ***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*********************** Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7125 Location: Pune, India Followers: 2134 Kudos [?]: 13653 [0], given: 222 Re: x+/x/+y=7 and x+/y/-y=6, then x+y=? [#permalink] ### Show Tags 02 Mar 2015, 19:56 amianik wrote: x+/x/+y=7 and x+/y/-y=6, then x+y=? a. -1 b. 1 c. 3 d. 5 e. 13 please explain this x + |x| + y = 7 x + |y| - y = 6 We can easily guess the signs of x and y. Consider the first equation: x + |x| + y = 7 If x is negative, |x| = -x and the equation reduces to y = 7. But if y = 7, second equation gives us x = 6 (not possible since we assumed x to be negative). So x must be positive. Consider second equation: x + |y| - y = 6 If y is positive, |y| = y and this equation reduces to x = 6. But if x = 6, first equation gives us y = -5 (not possible since we assumed y to be positive). So y must be negative. So equations become: x + x + y = 7 and x - y - y = 6 Solving them simultaneously, we get x = 4 and y = -1 so x+y = 3 Answer (C) _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: x+/x/+y=7 and x+/y/-y=6, then x+y=? [#permalink]

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02 Mar 2015, 21:25
VeritasPrepKarishma wrote:
amianik wrote:
x+/x/+y=7 and x+/y/-y=6, then x+y=?

a. -1
b. 1
c. 3
d. 5
e. 13

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

We can easily guess the signs of x and y.
Consider the first equation: x + |x| + y = 7
If x is negative, |x| = -x and the equation reduces to y = 7. But if y = 7, second equation gives us x = 6 (not possible since we assumed x to be negative). So x must be positive.
Consider second equation: x + |y| - y = 6
If y is positive, |y| = y and this equation reduces to x = 6. But if x = 6, first equation gives us y = -5 (not possible since we assumed y to be positive). So y must be negative.

So equations become: x + x + y = 7 and x - y - y = 6
Solving them simultaneously, we get x = 4 and y = -1 so x+y = 3

what is the reason to take x=negative arbitrary?is it due to cancel ou x on the equation and we will get y value?? if i suppose x= positive then how we can prove that y will be negative??
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink]

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03 Mar 2015, 04:27
amianik wrote:
x+/x/+y=7 and x+/y/-y=6, then x+y=?

a. -1
b. 1
c. 3
d. 5
e. 13

Merging topics. Please refer to the discussion on pages 1 and 2.
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink]

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03 Mar 2015, 21:24
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Expert's post
amianik wrote:
VeritasPrepKarishma wrote:
amianik wrote:
x+/x/+y=7 and x+/y/-y=6, then x+y=?

a. -1
b. 1
c. 3
d. 5
e. 13

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

We can easily guess the signs of x and y.
Consider the first equation: x + |x| + y = 7
If x is negative, |x| = -x and the equation reduces to y = 7. But if y = 7, second equation gives us x = 6 (not possible since we assumed x to be negative). So x must be positive.
Consider second equation: x + |y| - y = 6
If y is positive, |y| = y and this equation reduces to x = 6. But if x = 6, first equation gives us y = -5 (not possible since we assumed y to be positive). So y must be negative.

So equations become: x + x + y = 7 and x - y - y = 6
Solving them simultaneously, we get x = 4 and y = -1 so x+y = 3

what is the reason to take x=negative arbitrary?is it due to cancel ou x on the equation and we will get y value?? if i suppose x= positive then how we can prove that y will be negative??

To get rid of |x|, you will need to assume two cases: x positive and x negative. We see that if |x| = -x, then the equation simplifies greatly. So quickly analyze that case first and you see that it is not possible. So you are left with only 'x is positive' to consider. You are, in effect, considering both cases but ruling out one quickly by just looking at the equations.
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink]

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12 Jan 2017, 01:43
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

FAQ: If y is negative shouldn't it be x+2y = 6?

The basic idea is that a variable can represent a negative value, and if it does, then any subtraction or addition in an equation is therefore reversed, in a way.

Say, for example, that y = -1

In the expression x - y, then, we are NOT looking at x - 1. Instead, we have x - (-1), which is x + 1. Even though we are subtracting y (as in x - y), we are increasing the value (as in x + 1).

Similarly, in x - 2y, we would have x + 2.

So in the equation x + |y| - y, we are increasing the value twice: once with "+ |y|" and once with "- y".

So to simplify that, we subtract 2y from x, because y is negative, so we're actually increasing the value.

In case this still isn't clear, let's break it down as much as possible and compare the two equations using y = -1:

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

Now let's compare this result with the equation x - 2y = 6,

x - 2y = 6
x - (2 * -1) = 6
x - (-2) = 6
x + 2 = 6

See how we start with a negative and end up with the same result?

FAQ: I still don't understand why it's not x + 2y = 6. Could you try explaining this in another way?

Sure, let's try a substitution game! We have a variable y that must represent a negative number:

y = neg

Let's now replace "y" with "neg" in our equation:

x + |y| - y = 6
x + |neg| - neg = 6

The absolute value of that negative number must result in the positive version of that number. Likewise, subtracting that negative number must result in the positive version of that number. So we end up with the following where "pos" represents the positive version of that number:

x + pos + pos = 6
x + 2 * pos = 6

But... how do we get this equation back in terms of y? We know that y cannot be a positive number since we explicitly declared that y must be a negative number. Thus, y cannot equal pos. So, is there some way to relate y to pos?

Yes! The negative equivalent to any positive number is just that positive number times -1:

pos * -1 = neg

And since y = neg, we can replace neg with y:

pos * -1 = y
pos * -1 / -1 = y / -1
pos = -y

Now that we have pos in terms of y, we can make a final substitution in our equation:

x + 2 * pos = 6
x + 2 * (-y) = 6
x + -2y = 6
x - 2y = 6
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