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Senior Manager  Joined: 13 May 2013
Posts: 389
Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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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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1
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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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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1
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]

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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

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

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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).

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]

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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).

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 $$x+2-(-2)=6$$ --> $$x+4=6$$ ($$x-2y=6$$).

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

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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

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

Originally posted by gkashyap on 30 Mar 2014, 07:32.
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]

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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]

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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]

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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

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

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

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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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1
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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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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If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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Is there another way to solve this , perhaps number line method -- any suggestions experts ?

solving brute force, you have to test 9 scenarios which will easily take more than 3 mins...

Whats the fastest way to solve this if not brute force ?

Test cases when using brute force, i am counting 9 test cases !!
a) x = 0 | y = 0
b) x >0 | y >0
c) x <0 | y <0
d) x <0 | y>0
e) x >0 | y <0
f) x = 0 | y >0
g) x = 0 | y <0
i) x >0 | y =0
j) x <0 | y = 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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jabhatta@umail.iu.edu wrote:
Is there another way to solve this , perhaps number line method -- any suggestions experts ?

solving brute force, you have to test 9 scenarios which will easily take more than 3 mins...

Whats the fastest way to solve this if not brute force ?

Test cases when using brute force, i am counting 9 test cases !!
a) x = 0 | y = 0
b) x >0 | y >0
c) x <0 | y <0
d) x <0 | y>0
e) x >0 | y <0
f) x = 0 | y >0
g) x = 0 | y <0
i) x >0 | y =0
j) x <0 | y = 0

Check here for a faster method: https://gmatclub.com/forum/if-x-x-y-7-a ... l#p1492946
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If x+|x|+y=7 and x+|y|-y=6 what is x+y=?  [#permalink]

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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

Hi - I did the following but got a different result ....Please let me know why are we getting different result

So I started testing the following

-- Assumed Y was negative ...

-- Hence in equation 2, x = 6

-- Hence checked if x is positive in equation 1

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

hence y = -5

This went with my assumption that Y had to be negative (specifically negative 5) and X was positive (specifically positive 6)

Furthermore, plugging X = 6 and Y = -5 ...works in both equations

X + Y = 6 - 5 = 1

Hence i chose B

Please let me know why cant X = 6 and Y = -5 be the actual values of X and 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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jabhatta@umail.iu.edu 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

Hi - I did the following but got a different result ....Please let me know why are we getting different result

So I started testing the following

-- Assumed Y was negative ...

-- Hence in equation 2, x = 6

-- Hence checked if x is positive in equation 1

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

hence y = -5

This went with my assumption that Y had to be negative (specifically negative 5) and X was positive (specifically positive 6)

Furthermore, plugging X = 6 and Y = -5 ...works in both equations

X + Y = 6 - 5 = 1

Hence i chose B

Please let me know why cant X = 6 and Y = -5 be the actual values of X and Y ..

hi..

you are wrong in coloured portion and since it is right in beginning, the entire answer goes wrong..

One way is to check combinations tha tBOTh are negative, or both are positive or both are different.
One of the above cases would be y as negative and x as positive, what you have taken

if y is negative
x + |y| - y = 6 will become x-2y=6.................. -2y is POSITIVE as y is negative
lets take second equation..
x + |x| + y = 7 will become 2x+y=7........y=7-2x

substitute in x-2y=6 .......
x-2(7-2x)=6.......x-14+4x=6......5x=14+6=20.....x=4
y=7-2x=7-2*4=-1..

so it matches with the assumption we started with that x is positive and y is negative..

ans = x+y=4+(-1)=4-1=3
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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Hi All,

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

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

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_________________ Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?   [#permalink] 03 Mar 2020, 14:07

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