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# If 6xy = x^2y + 9y, what is the value of xy?

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If 6xy = x^2y + 9y, what is the value of xy? [#permalink]

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19 Dec 2010, 16:04
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If 6xy = $$x^2$$y + 9y, what is the value of xy?

(1) x = –2
(2) x < 0

[Reveal] Spoiler:
The answer is D, with the following explanation:
The equation in question can be rephrased as follows:

x2y – 6xy + 9y = 0
y(x2 – 6x + 9) = 0
y(x – 3)2 = 0

Therefore, one or both of the following must be true:
y = 0 or
x = 3

It follows that the product xy must equal either 0 or 3y. This question can therefore be rephrased several different ways: "What is y?" or "Does x = 3?"

(1) SUFFICIENT: If x = -2, then:
(–2)2y – 6(–2)y + 9y = 0
4y + 12y + 9y = 0
25y = 0

Therefore y = xy = 0. Tn fact, for any value of x other than 3, y must equal zero.

(2) SUFFICIENT: x is negative. Therefore, x cannot equal 3, and it follows that y = 0. Therefore, xy = 0.

What I did, i simply applied the info from statement 1, and got that -12Y=13Y which can only be true if Y is 0 and therefore, XY is 0.
In the second statement, i did the same and the result came to be the same. Is this faulty reasoning? Also, this is said to be 700-800 level question and my way seems just too easy to solve such a high level problem...
[Reveal] Spoiler: OA

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http://gmatclub.com/forum/massive-collection-of-verbal-questions-sc-rc-and-cr-106195.html#p832142

[highlight]Massive collection of thousands of Data Sufficiency and Problem Solving questions and answers:[/highlight]
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Last edited by reto on 16 Jul 2015, 00:55, edited 1 time in total.
proper format

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Re: If 6xy = x^2y + 9y, what is the value of xy? [#permalink]

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19 Dec 2010, 16:22
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MisterEko wrote:
Guys, I just have a question if my way of solving it is reasonable and can often be applied here. I got the right answer on this one, but am not sure it would work usually:

If 6xy = x2y + 9y, what is the value of xy?

(1) x = –2

(2) x < 0

The answer is D, with the following explanation:
[Reveal] Spoiler:
The equation in question can be rephrased as follows:

x2y – 6xy + 9y = 0
y(x2 – 6x + 9) = 0
y(x – 3)2 = 0

Therefore, one or both of the following must be true:
y = 0 or
x = 3

It follows that the product xy must equal either 0 or 3y. This question can therefore be rephrased several different ways: "What is y?" or "Does x = 3?"

(1) SUFFICIENT: If x = -2, then:
(–2)2y – 6(–2)y + 9y = 0
4y + 12y + 9y = 0
25y = 0

Therefore y = xy = 0. Tn fact, for any value of x other than 3, y must equal zero.

(2) SUFFICIENT: x is negative. Therefore, x cannot equal 3, and it follows that y = 0. Therefore, xy = 0.

What I did, i simply applied the info from statement 1, and got that -12Y=13Y which can only be true if Y is 0 and therefore, XY is 0.
In the second statement, i did the same and the result came to be the same. Is this faulty reasoning? Also, this is said to be 700-800 level question and my way seems just too easy to solve such a high level problem...

For statement (1) your reasoning looks OK. As for statement (2) what do you mean by "did the same"?

One can also do this way: $$6xy=x^2y + 9y$$ --> $$y(x^2-6x+9)=0$$ --> $$y(x-3)^2=0$$ --> either $$x=3$$ or $$y=0$$ (or both).

(1) x = –2 --> so $$x\neq{3}$$, then $$y=0$$ and $$xy=0$$. Sufficient.

(2) x < 0 --> again $$x\neq{3}$$, then $$y=0$$ and $$xy=0$$. Sufficient.

Check question #1 here: inequality-and-absolute-value-questions-from-my-collection-86939.html for harder version of the same problem.

Hope it helps.
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Re: If 6xy = x^2y + 9y, what is the value of xy? [#permalink]

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19 Dec 2010, 17:01
Bunuel wrote:
MisterEko wrote:
Guys, I just have a question if my way of solving it is reasonable and can often be applied here. I got the right answer on this one, but am not sure it would work usually:

If 6xy = x2y + 9y, what is the value of xy?

(1) x = –2

(2) x < 0

The answer is D, with the following explanation:
[Reveal] Spoiler:
The equation in question can be rephrased as follows:

x2y – 6xy + 9y = 0
y(x2 – 6x + 9) = 0
y(x – 3)2 = 0

Therefore, one or both of the following must be true:
y = 0 or
x = 3

It follows that the product xy must equal either 0 or 3y. This question can therefore be rephrased several different ways: "What is y?" or "Does x = 3?"

(1) SUFFICIENT: If x = -2, then:
(–2)2y – 6(–2)y + 9y = 0
4y + 12y + 9y = 0
25y = 0

Therefore y = xy = 0. Tn fact, for any value of x other than 3, y must equal zero.

(2) SUFFICIENT: x is negative. Therefore, x cannot equal 3, and it follows that y = 0. Therefore, xy = 0.

What I did, i simply applied the info from statement 1, and got that -12Y=13Y which can only be true if Y is 0 and therefore, XY is 0.
In the second statement, i did the same and the result came to be the same. Is this faulty reasoning? Also, this is said to be 700-800 level question and my way seems just too easy to solve such a high level problem...

For statement (1) your reasoning looks OK. As for statement (2) what do you mean by "did the same"?

One can also do this way: $$6xy=x^2y + 9y$$ --> $$y(x^2-6x+9)=0$$ --> $$y(x-3)^2=0$$ --> either $$x=3$$ or $$y=0$$ (or both).

(1) x = –2 --> so $$x\neq{3}$$, then $$y=0$$ and $$xy=0$$. Sufficient.

(2) x < 0 --> again $$x\neq{3}$$, then $$y=0$$ and $$xy=0$$. Sufficient.

Check question #1 here: inequality-and-absolute-value-questions-from-my-collection-86939.html for harder version of the same problem.

Hope it helps.

Thanks a lot. What I meant by "I did the same" is that I substituted x for -1 and tested, and result came out to be the same...
_________________

[highlight]Monster collection of Verbal questions (RC, CR, and SC)[/highlight]
http://gmatclub.com/forum/massive-collection-of-verbal-questions-sc-rc-and-cr-106195.html#p832142

[highlight]Massive collection of thousands of Data Sufficiency and Problem Solving questions and answers:[/highlight]
http://gmatclub.com/forum/1001-ds-questions-file-106193.html#p832133

Kudos [?]: 475 [0], given: 36

Math Expert
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Kudos [?]: 139431 [0], given: 12790

Re: If 6xy = x^2y + 9y, what is the value of xy? [#permalink]

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19 Dec 2010, 23:53
MisterEko wrote:
Thanks a lot. What I meant by "I did the same" is that I substituted x for -1 and tested, and result came out to be the same...

Yes, for ANY x but 3 you'll get ky=0 for some nonzero k and thus y=0 (for x=3 you'll get 0=0).
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Re: If 6xy = x^2y + 9y, what is the value of xy? [#permalink]

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11 Dec 2013, 03:14
1
This post was
BOOKMARKED
Bunuel wrote:
MisterEko wrote:
Guys, I just have a question if my way of solving it is reasonable and can often be applied here. I got the right answer on this one, but am not sure it would work usually:

If 6xy = x2y + 9y, what is the value of xy?

(1) x = –2

(2) x < 0

The answer is D, with the following explanation:
[Reveal] Spoiler:
The equation in question can be rephrased as follows:

x2y – 6xy + 9y = 0
y(x2 – 6x + 9) = 0
y(x – 3)2 = 0
Therefore, one or both of the following must be true:
y = 0 or
x = 3

It follows that the product xy must equal either 0 or 3y. This question can therefore be rephrased several different ways: "What is y?" or "Does x = 3?"

(1) SUFFICIENT: If x = -2, then:
(–2)2y – 6(–2)y + 9y = 0
4y + 12y + 9y = 0
25y = 0

Therefore y = xy = 0. Tn fact, for any value of x other than 3, y must equal zero.

(2) SUFFICIENT: x is negative. Therefore, x cannot equal 3, and it follows that y = 0. Therefore, xy = 0.

What I did, i simply applied the info from statement 1, and got that -12Y=13Y which can only be true if Y is 0 and therefore, XY is 0.
In the second statement, i did the same and the result came to be the same. Is this faulty reasoning? Also, this is said to be 700-800 level question and my way seems just too easy to solve such a high level problem...

For statement (1) your reasoning looks OK. As for statement (2) what do you mean by "did the same"?

One can also do this way: $$6xy=x^2y + 9y$$ --> $$y(x^2-6x+9)=0$$ --> $$y(x-3)^2=0$$ --> either $$x=3$$ or $$y=0$$ (or both).

(1) x = –2 --> so $$x\neq{3}$$, then $$y=0$$ and $$xy=0$$. Sufficient.

(2) x < 0 --> again $$x\neq{3}$$, then $$y=0$$ and $$xy=0$$. Sufficient.

Check question #1 here: inequality-and-absolute-value-questions-from-my-collection-86939.html for harder version of the same problem.

Hope it helps.

Hi Bunuel,

I am still not clear why do we need statements. Once we know that y=0 then no matter what the value of x is xy will always be =0. Do we need to even consider the statements. Am I thinking in the right direction or is there a disconnect somewhere.

Kudos [?]: 44 [0], given: 15

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Re: If 6xy = x^2y + 9y, what is the value of xy? [#permalink]

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11 Dec 2013, 03:31
1
KUDOS
Expert's post
davidfrank wrote:
Bunuel wrote:
MisterEko wrote:
Guys, I just have a question if my way of solving it is reasonable and can often be applied here. I got the right answer on this one, but am not sure it would work usually:

If 6xy = x2y + 9y, what is the value of xy?

(1) x = –2

(2) x < 0

The answer is D, with the following explanation:
[Reveal] Spoiler:
The equation in question can be rephrased as follows:

x2y – 6xy + 9y = 0
y(x2 – 6x + 9) = 0
y(x – 3)2 = 0
Therefore, one or both of the following must be true:
y = 0 or
x = 3

It follows that the product xy must equal either 0 or 3y. This question can therefore be rephrased several different ways: "What is y?" or "Does x = 3?"

(1) SUFFICIENT: If x = -2, then:
(–2)2y – 6(–2)y + 9y = 0
4y + 12y + 9y = 0
25y = 0

Therefore y = xy = 0. Tn fact, for any value of x other than 3, y must equal zero.

(2) SUFFICIENT: x is negative. Therefore, x cannot equal 3, and it follows that y = 0. Therefore, xy = 0.

What I did, i simply applied the info from statement 1, and got that -12Y=13Y which can only be true if Y is 0 and therefore, XY is 0.
In the second statement, i did the same and the result came to be the same. Is this faulty reasoning? Also, this is said to be 700-800 level question and my way seems just too easy to solve such a high level problem...

For statement (1) your reasoning looks OK. As for statement (2) what do you mean by "did the same"?

One can also do this way: $$6xy=x^2y + 9y$$ --> $$y(x^2-6x+9)=0$$ --> $$y(x-3)^2=0$$ --> either $$x=3$$ or $$y=0$$ (or both).

(1) x = –2 --> so $$x\neq{3}$$, then $$y=0$$ and $$xy=0$$. Sufficient.

(2) x < 0 --> again $$x\neq{3}$$, then $$y=0$$ and $$xy=0$$. Sufficient.

Check question #1 here: inequality-and-absolute-value-questions-from-my-collection-86939.html for harder version of the same problem.

Hope it helps.

Hi Bunuel,

I am still not clear why do we need statements. Once we know that y=0 then no matter what the value of x is xy will always be =0. Do we need to even consider the statements. Am I thinking in the right direction or is there a disconnect somewhere.

See the red part is not true. We don't know whether $$y=0$$. From $$y(x-3)^2=0$$ we have that $$x=3$$ OR $$y=0$$. Now, if $$x=3$$, then y can be any number, thus xy is not necessarily 0.

Hope it's clear.
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Re: If 6xy = x^2y + 9y, what is the value of xy? [#permalink]

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11 Dec 2013, 05:21
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MisterEko wrote:
If 6xy = x^2y + 9y, what is the value of xy?

(1) x = –2
(2) x < 0

Responding to a pm:

Why do we need the statements?

Given: $$6xy = x^2y + 9y$$
Given: $$y(x-3)^2 = 0$$

What we know is this: EITHER y is 0 OR x is 3. It is possible that both y is 0 and x is 3 but at least one of them MUST be true. This is all we know. We don't know whether y is 0 or whether x is 3 or both.

Question: what is xy?
I cannot say yet. All I know is that either y is 0 or x is 3.
If y is 0, xy will be 0.
If x is 3, then I don't know y which could be anything so xy could be anything.
So I cannot say what xy is.

(1) x = –2

This tells me that x is not 3. If x is not 3, y MUST be 0 because one of them has to be true. If y is 0, we know for sure that xy is 0. Sufficient.

(2) x < 0

This again tells me that x is not 3. If x is not 3, y MUST be 0 because one of them has to be true. If y is 0, we know for sure that xy is 0. Sufficient.

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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 18470 [1], given: 237 Manager Joined: 31 Jul 2014 Posts: 144 Kudos [?]: 57 [0], given: 373 GMAT 1: 630 Q48 V29 Re: If 6xy = x^2y + 9y, what is the value of xy? [#permalink] ### Show Tags 17 Sep 2015, 07:53 Bunuel wrote: MisterEko wrote: Guys, I just have a question if my way of solving it is reasonable and can often be applied here. I got the right answer on this one, but am not sure it would work usually: If 6xy = x2y + 9y, what is the value of xy? (1) x = –2 (2) x < 0 The answer is D, with the following explanation: [Reveal] Spoiler: The equation in question can be rephrased as follows: x2y – 6xy + 9y = 0 y(x2 – 6x + 9) = 0 y(x – 3)2 = 0 Therefore, one or both of the following must be true: y = 0 or x = 3 It follows that the product xy must equal either 0 or 3y. This question can therefore be rephrased several different ways: "What is y?" or "Does x = 3?" (1) SUFFICIENT: If x = -2, then: (–2)2y – 6(–2)y + 9y = 0 4y + 12y + 9y = 0 25y = 0 Therefore y = xy = 0. Tn fact, for any value of x other than 3, y must equal zero. (2) SUFFICIENT: x is negative. Therefore, x cannot equal 3, and it follows that y = 0. Therefore, xy = 0. The correct answer is D. What I did, i simply applied the info from statement 1, and got that -12Y=13Y which can only be true if Y is 0 and therefore, XY is 0. In the second statement, i did the same and the result came to be the same. Is this faulty reasoning? Also, this is said to be 700-800 level question and my way seems just too easy to solve such a high level problem... For statement (1) your reasoning looks OK. As for statement (2) what do you mean by "did the same"? One can also do this way: $$6xy=x^2y + 9y$$ --> $$y(x^2-6x+9)=0$$ --> $$y(x-3)^2=0$$ --> either $$x=3$$ or $$y=0$$ (or both). (1) x = –2 --> so $$x\neq{3}$$, then $$y=0$$ and $$xy=0$$. Sufficient. (2) x < 0 --> again $$x\neq{3}$$, then $$y=0$$ and $$xy=0$$. Sufficient. Answer: D. Check question #1 here: inequality-and-absolute-value-questions-from-my-collection-86939.html for harder version of the same problem. Hope it helps. why we can not cancel out Y from 6xy=x^2y + 9y ? this is not inequality..what is mistake in my thinking Kudos [?]: 57 [0], given: 373 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7866 Kudos [?]: 18470 [3], given: 237 Location: Pune, India Re: If 6xy = x^2y + 9y, what is the value of xy? [#permalink] ### Show Tags 17 Sep 2015, 19:24 3 This post received KUDOS Expert's post anupamadw wrote: why we can not cancel out Y from 6xy=x^2y + 9y ? this is not inequality..what is mistake in my thinking Usually, it is not a good idea to cancel out a variable. You lose a solution if you do. If you do not cancel out the y, you get that this inequality: 6xy=x^2y + 9y holds when either x = 3 or y = 0. If you do cancel out the y, you get that this inequality: 6xy=x^2y + 9y holds when x = 3. You lost the y = 0 value. Hence, it's always better to take out y common and keep it on the side, not cancel it. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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If 6xy = x^2y + 9y, what is the value of xy? [#permalink]

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31 Dec 2017, 05:42
[quote="MisterEko"]If 6xy = $$x^2$$y + 9y, what is the value of xy?

(1) x = –2
(2) x < 0

[spoiler=]The answer is D, with the following explanation:
The equation in question can be rephrased as follows:

Slightly different analysis. Bunuel / Experts pl. validate

Given $$6xy=x^2y+9y$$

Find xy= ?

Now $$6xy=x^2y+9y$$ ------ (1)
=> $$6xy=y(x^2+9)$$
=> $$6x=x^2+9$$
=> $$x^2-6x+9=0$$
=> $$x^2-3x-3x+9=0$$
=> $$(x-3)^2=0$$
=> Therefore x=3
=> Substituting x=3 in (1) we have
=> 0=0

So we know the value of 'x'. We need to find y=?

Statement 1 x=-2
=> Substituting in (1) we have
=> -12y=13y
=> OR 25y=0
=> OR y=0
=> Therefore xy=0 SUFFICIENT

Statement 2 x<0 i.e 'x' is -ve
=> Substituting value of 'x' in equ (1)
=> Now irrespective of the magnitude ' -x ' LHS of equ (1) will be -ve
=> i.e 6xy= -ky ---- for some constant 'k'
=> Again irrespective of the magnitude ' -x ' RHS of equ (1) will be +ve
=> i.e $$x^2y+9y=my$$ ---- for some constant 'm'
=> -ky=my
=> OR my+ky=0
=> OR ny=0 ----- for some constant 'n'
=> Irrespective of the SIGN of n, y=0
=> Therefore xy=0. SUFFICIENT

Therefore "D"

Thanks
Dinesh

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If 6xy = x^2y + 9y, what is the value of xy?   [#permalink] 31 Dec 2017, 05:42
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