If sqrt(xy)=xy , what is the value of x + y? (1) x = -1/2 : GMAT Data Sufficiency (DS)
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# If sqrt(xy)=xy , what is the value of x + y? (1) x = -1/2

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If sqrt(xy)=xy , what is the value of x + y? (1) x = -1/2 [#permalink]

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05 Sep 2009, 18:30
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If sqrt(xy)=xy , what is the value of x + y?

(1) x = -1/2
(2) y is not equal to zero
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05 Sep 2009, 19:08
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If $$\sqrt{xy} = xy$$ what is the value of x + y?

$$\sqrt{xy} = xy$$ --> $$xy=x^2y^2$$ --> $$x^2y^2-xy=0$$ --> $$xy(xy-1)=0$$ --> either $$xy=0$$ or $$xy=1$$.

(1) x = -1/2 --> either $$-\frac{1}{2}*y=0$$ --> $$y=0$$ and $$x+y=-\frac{1}{2}$$ OR $$-\frac{1}{2}*y=1$$ --> $$y=-2$$ and $$x+y=-\frac{5}{2}$$. Not sufficient.

(2) y is not equal to zero. Clearly not sufficient.

(1)+(2) Since from (2) $$y\neq{0}$$, then from (1) $$y=-2$$ and $$x+y=-\frac{5}{2}$$. Sufficient.

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Last edited by Bunuel on 05 Sep 2009, 20:07, edited 1 time in total.
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05 Sep 2009, 19:58
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now we can write eq as:-
[square_root]xy=xy...... xy=(xy)^2.....ie (xy)^2-xy=0.....or xy(xy-1)=0....
so xy=0 or xy=1
i)x=-1/2.....
substituting this value in xy we get (-1/2)y=0 ....so y=0...
also (-1/2)y=0....y=-2.... not sufficient....
ii)y not equal to 0.... not sufficient...
combining the two.... y=-2... sufficient
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06 Sep 2009, 03:12
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[quote="dhushan"]If sqrt(xy)=xy , what is the value of x + y?
(1) x = -1/2
(2) y is not equal to zero

XY=(XY)^2........ie: x,y have the same sign and they could be (0,anything)(1,1),(-1,-1) receprocals

from 1

no info about y......x,y could be (0,-1/2) or receprocals
from 2
insuff

both

receprocals..........suff
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06 Sep 2009, 05:44
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Thanks guys, I see how your answers work, but I was wondering what is wrong with solving the problem this way.

sqrt(xy) = xy

xy = x^2y^2
1/x = y

therefore if y = 1/x, and from the info in (1), couldn't we deduce that y = =-2 by substituting -1/2 in for x?

Therefore A would be the answer?
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06 Sep 2009, 05:53
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dhushan wrote:
Thanks guys, I see how your answers work, but I was wondering what is wrong with solving the problem this way.

sqrt(xy) = xy

xy = x^2y^2
1/x = y

therefore if y = 1/x, and from the info in (1), couldn't we deduce that y = =-2 by substituting -1/2 in for x?

Therefore A would be the answer?

You cancelled out a root of the equation which is incorrect..you should consider each and every real root of the equation..
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06 Sep 2009, 06:07
gmate2010 wrote:
dhushan wrote:
Thanks guys, I see how your answers work, but I was wondering what is wrong with solving the problem this way.

sqrt(xy) = xy

xy = x^2y^2
1/x = y

therefore if y = 1/x, and from the info in (1), couldn't we deduce that y = =-2 by substituting -1/2 in for x?

Therefore A would be the answer?

You cancelled out a root of the equation which is incorrect..you should consider each and every real root of the equation..

Sorry, I still don't follow. what do you mean by "cancelled out a root of the equation" - I still have x and y in the equation.
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06 Sep 2009, 07:54
for a function the square root of xy is only equal to xy if the function is equal to 0 or 1, you can do the math and find the roots by squaring but I just accept that it can only equal 0 or 1. So if we know X is not 0 and is in fact a #, we know Y can only be 0 or the multiplicative reciporcal of X so that XY=1 or 0. When we get statement 2 we know that X*Y can not be equal to 0 so we know that XY= 1 and if we know what X is we can calculate what Y is.

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06 Sep 2009, 09:38
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dhushan wrote:
Thanks guys, I see how your answers work, but I was wondering what is wrong with solving the problem this way.

sqrt(xy) = xy

xy = x^2y^2
1/x = y

therefore if y = 1/x, and from the info in (1), couldn't we deduce that y = =-2 by substituting -1/2 in for x?

Therefore A would be the answer?

x,y are both interrelated ( roots ie: the solution is a unique combination of x,y value.s but not any of them on its own),

you can never cancell out a VARIABLE, because its unique value in the combination xy or x^2y^2 makes the equation valid.

think of it as if there are 2 conditions for the equation to be true the first is the value of x and the second is values of y but not any of them alone.

hope am clear
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06 Sep 2009, 12:42
yezz wrote:
dhushan wrote:
Thanks guys, I see how your answers work, but I was wondering what is wrong with solving the problem this way.

sqrt(xy) = xy

xy = x^2y^2
1/x = y

therefore if y = 1/x, and from the info in (1), couldn't we deduce that y = =-2 by substituting -1/2 in for x?

Therefore A would be the answer?

x,y are both interrelated ( roots ie: the solution is a unique combination of x,y value.s but not any of them on its own),

you can never cancell out a VARIABLE, because its unique value in the combination xy or x^2y^2 makes the equation valid.

think of it as if there are 2 conditions for the equation to be true the first is the value of x and the second is values of y but not any of them alone.

hope am clear

Thanks, I finally get it in this situation. However, does the same hold true in other questions, for example

x^2y^2 = x^2 (so here I would have determine whether, x = 0 and y = 0, or x and y = 1)

Thanks for everyone's help, it is greatly appreciated.
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06 Sep 2009, 13:11
Here you'll have:

x^2y^2 -x^2 = 0 --> x^2(y^2-1)=0 --> x^2(y-1)(y+1)=0

One of the multiples must be zero --> x=0, y=1 or y=-1.
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28 Sep 2009, 03:11
dhushan wrote:
yezz wrote:
dhushan wrote:
Thanks guys, I see how your answers work, but I was wondering what is wrong with solving the problem this way.

sqrt(xy) = xy

xy = x^2y^2
1/x = y

therefore if y = 1/x, and from the info in (1), couldn't we deduce that y = =-2 by substituting -1/2 in for x?

Therefore A would be the answer?

x,y are both interrelated ( roots ie: the solution is a unique combination of x,y value.s but not any of them on its own),

you can never cancell out a VARIABLE, because its unique value in the combination xy or x^2y^2 makes the equation valid.

think of it as if there are 2 conditions for the equation to be true the first is the value of x and the second is values of y but not any of them alone.

hope am clear

Thanks, I finally get it in this situation. However, does the same hold true in other questions, for example

x^2y^2 = x^2 (so here I would have determine whether, x = 0 and y = 0, or x and y = 1)

Thanks for everyone's help, it is greatly appreciated.

Here is the catch .....

In mathematics ... division by zero is not allowed ...
so
xy = x^2Y^2 => x^2y^2 - xy = 0 => xy(xy - 1) = => xy = 0 or xy = 1 => x is not zero therefor y = 0 or y =1/x

in ur second case .....

x^2y^2 = x^2 => x^2y^2 - x^2 = 0 => x^2(y^2 - 1) = 0 => x^2 = 0 or y^2 = 1 => x = 0 or y = 1 or y = -1...
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28 Sep 2009, 03:47
statement 1:
==========
x = -1/2 .so y can be 0 or -2.Nt suff

Statement 2:
==========
y is not equal to zero. Nt suff

Combining both we can get x = -1/2 y = -2.
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28 Sep 2009, 04:51
dhushan wrote:
If sqrt(xy)=xy , what is the value of x + y?
(1) x = -1/2
(2) y is not equal to zero

sqrt(xy)=xy -> two solutions: xy = 0 or xy = 1.

1: insufficient: y = -2 or y = 0
2: insufficient: x = 1/y

1+2: sufficient y = -2, x = -1/2 -> x+y = -2.5 -> C
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29 Sep 2009, 09:18
Hey guys - Just a quick clarification

Why can't we divide the equation by \sqrt{xy} to yield the following:

1 = \sqrt{xy}

Based on this .. just option 1 would be sufficient because if x is -1/2 y has to be -2 to satisfy this above equation.
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29 Sep 2009, 18:16
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deepakraam wrote:
statement 1:
==========
x = -1/2 .so y can be 0 or -2.Nt suff

Statement 2:
==========
y is not equal to zero. Nt suff

Combining both we can get x = -1/2 y = -2.

Simple and clear solution
thanku
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Re: If sqrt(xy)=xy , what is the value of x + y? (1) x = -1/2 [#permalink]

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22 Dec 2015, 09:41
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If sqrt(xy)=xy , what is the value of x + y? (1) x = -1/2 [#permalink]

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09 Mar 2016, 17:52
Bunuel wrote:
If $$\sqrt{xy} = xy$$ what is the value of x + y?

$$\sqrt{xy} = xy$$ --> $$xy=x^2y^2$$ --> $$x^2y^2-xy=0$$ --> $$xy(xy-1)=0$$ --> either $$xy=0$$ or $$xy=1$$.

(1) x = -1/2 --> either $$-\frac{1}{2}*y=0$$ --> $$y=0$$ and $$x+y=-\frac{1}{2}$$ OR $$-\frac{1}{2}*y=1$$ --> $$y=-2$$ and $$x+y=-\frac{5}{2}$$. Not sufficient.

(2) y is not equal to zero. Clearly not sufficient.

(1)+(2) Since from (2) $$y\neq{0}$$, then from (1) $$y=-2$$ and $$x+y=-\frac{5}{2}$$. Sufficient.

On combining both the statements, we still wouldn't know the value of x. What if x is 0?

Last edited by atirajak on 09 Mar 2016, 21:08, edited 1 time in total.
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Re: If sqrt(xy)=xy , what is the value of x + y? (1) x = -1/2 [#permalink]

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09 Mar 2016, 18:12
atirajak wrote:
Bunuel wrote:
If $$\sqrt{xy} = xy$$ what is the value of x + y?

$$\sqrt{xy} = xy$$ --> $$xy=x^2y^2$$ --> $$x^2y^2-xy=0$$ --> $$xy(xy-1)=0$$ --> either $$xy=0$$ or $$xy=1$$.

(1) x = -1/2 --> either $$-\frac{1}{2}*y=0$$ --> $$y=0$$ and $$x+y=-\frac{1}{2}$$ OR $$-\frac{1}{2}*y=1$$ --> $$y=-2$$ and $$x+y=-\frac{5}{2}$$. Not sufficient.

(2) y is not equal to zero. Clearly not sufficient.

(1)+(2) Since from (2) $$y\neq{0}$$, then from (1) $$y=-2$$ and $$x+y=-\frac{5}{2}$$. Sufficient.

What if x is 0? On combining both the statements, we still wouldn't know the value of x.

Your question is confusing. On one hand you are assuming that x=0 and on the other hand you are saying that you dont know the value of x. Can you rephrase your question?

As for this question, when you combine both the statements, x=-0.5 and as y $$\neq$$ 0 ---> this thus rules out the case when xy=0, leaving you with unique values of y and x. Hence C is the correct answer.

Also, S2 alone does leave the door open for assuming x=0 but then again it can very well be $$\neq$$ 0, making statement 2 not sufficient.

Hope this helps.
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Re: If sqrt(xy)=xy , what is the value of x + y? (1) x = -1/2   [#permalink] 09 Mar 2016, 18:12

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