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

Question

(1) y – x = 3

(2) x^3 < 0

Manhnip · Accepted Answer

The equation in question can be rephrased as follows: 
x^2 y – 6xy + 9y = 0

y(x^2 –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 “What is y?”

(1) INSUFFICIENT: This equation cannot be manipulated or combined with the original equation to solve directly for x or y. Instead, plug the two possible scenarios from the original equation into the equation from this statement:

If x = 3, then y = 3 + x = 3 + 3 = 6, so xy = (3)(6) = 18. If y = 0, then x = y – 3 = 0 – 3 = -3, so xy = (-3)(0) = 0.
Since there are two possible answers, this statement is not sufficient.

(2) SUFFICIENT: If x3 < 0, then x < 0. Therefore, x cannot equal 3, and it follows that y = 0. Therefore, xy = 0.

Ans B

Please someone confirm on the solution, if the approach is correct

Bunuel · Answer

Yes, that's correct. If 6xy = x^2y + 9y, what is the value of xy? 6xy=x^2y + 9y y(x^2-6x+9)=0 ; y(x-3)^2=0 . Either x=3 or y=0 (or both). (1) y – x = 3. If y=0 and x=-3 , then xy=0 . However, if x=3 and y=6 , then xy=18 . Not sufficient. (2) x^3 < 0. The above implies that x<0 . Hence, x eq{3} , and thus y=0 and xy=0 . Sufficient. Answer: B. Similar question to practice: https://gmatclub.com/forum/if-6xy-x-2y- ... 06556.html Hope it helps.

Himalayan · Answer

if (x2y) in (6xy = x2y + 9y) is x^2y, x can not be -ve.

6xy = x^2 + 9y
6xy = y (x^2 + 9)
x^2 - 6x + 9 = 0
(x - 3)^2 = 0
so x = 3.

1: y = 6. sufficient.
2: absurd. nsf.

probably A but the question is not a formulated properly.

himanshujovi · Answer

Aaah this formatting.. I read the stem as x^(2y) and was totally stuck. Can this ambiguity in notation come in actual test ? I would have expected a bracket or at the least correct ordering like 6xy = yx^2

Bunuel · Answer

Edited the original post to avoid such confusions in future.

By the way x^2y means x^2*y. If it were  x^{2y}  it would be written x^(2y). But don't worry on the actual test the formatting will be clearer.

Hope it helps.

ravih · Answer

Bunnuel,

Why is it wrong to divide the original equation by y? Since dividing the equation by y gives you x^2-6x+9=0 which gives you x=3, then statement 1 is sufficient to give you the value of y and therefore give you the value of xy as 18.

Statement 2 is not consistent with the original equation as it shows x^3<0 which is not possible (x=3, so 3^3=27) and is therefore insufficient. 
Therefore, answer should be A.

Or is it that only because statement 2 is inconsistent with the original equation and therefore we need to try a different approach?

Bunuel · Answer

If you divide (reduce) 6xy = x^2*y + 9y by y, you assume, with no ground for it, that y does not equal to zero thus exclude a possible solution (notice that both y = 0 AND x = 3 satisfy the equation).

Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We can not divide by zero.

ravih · Answer

Thats an important lesson! thanks Bunuel!

mais1990 · Answer

Bunuel, I did but there is some minor point that I am just not getting. For instance: 
if x3 < 0 i.e. x is negative, how do we know that y will be 0?
I am not following how either the original equation - y(x+3)^2 =0 - or the above solutions are leading us to this conclusion. 
thanks a ton

Bunuel · Answer

For y(x−3)^2 to be 0, either x must be 3 or y must be 0 (or both). x^3 < 0 implies that x is NOT 3, thus y MUST be 0 (in order y(x−3)^2 to be 0).

mais1990 · Answer

Now, I get it. Thank you!

MathRevolution · Answer

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

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

(1) y – x = 3

(2) x^3 < 0

If we modify the question, 0=x^2*y-6xy+9y, or 0=y(x^2-6x+9), or 0=y(x-3)^2, and we want to know whether y=0 or x=3. There are 2 variables (x,y) and 1 equation (0=x^2*y-6xy+9y). 2 more equations are given from the 2 conditions, so there is high chance (D) will be our answer.
From condition 1, y=0 and x=-3 or x=3 and y=6. This is not sufficient as this is not unique.
From condition 2, x^3<0--> x<0. But x=3>0, and y has to be y=0, and xy=0, so the condition is sufficient; the answer is (B).

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.

SUNILAA · Answer

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
“What is y?”

1)  INSUFFICIENT : This equation cannot be manipulated or combined with the original equation to
solve directly for x or y. Instead, plug the two possible scenarios from the original equation into the
equation from this statement:
If x = 3, then y = 3 + x = 3 + 3 = 6, so xy = (3)(6) = 18.
If y = 0, then x = y – 3 = 0 – 3 = -3, so xy = (-3)(0) = 0.
Since there are two possible answers, this statement is not sufficient.
(2) SUFFICIENT: If x3 < 0, then x < 0. Therefore, x cannot equal 3, and it follows that y = 0.
Therefore, xy = 0.
The correct answer is B.

GMATinsight · Answer

This video is not only the video solution of this question, but also an explanation how the DS questions should be approached.

It also highlights a common mistake of many test takers

https://www.youtube.com/watch?v=kszOy_LCh2s

Rajashree123 · Answer

it is given in the statement x=+- 3,y=0.
then in statement 1, why we are taking different value (y=6) . Is it to satisfy the equation?

bcz as per my ans D should be correct.

Sambon · Answer

Bunuel's approach is the best, but here is an alternative solution.
Rearrange the equation in the stem to  xy = \frac{y(x^2+9)}{6}  and now you're looking for the value of xy

(1)  y-x=3  so  y=3+x  > plug this into the equation in the stem
 x(3+x)=\frac{(3+x)(x^2+9)}{6} 
 3x+x^2 = \frac{3x^2+27+x^3+9x}{6} 
 18x+6x^2=3x^2+27+x^3+9x 
 x^3-3x^2-9x+27=0 
 x(x^2-9)-3(x^2-9)=0 
 (x+3)(x-3)^2=0 
x=3, -3
If x=3, y=6 and xy=18. If x=-3, y=0 and xy=0. NOT SUFFICIENT.

(2)  x^3<0 
Notice that if x is negative then y must be equal to zero because in the equation  xy = \frac{y(x^2+9)}{6}  x will be negative on the LHS while (x^2+9) will be positive. Since a negative cannot equal a positive, y must equal zero to make the equation valid. SUFFICIENT

namanqqkm · Answer

why can't this equation, 6xy = x2y + 9y, be written as 
6xy -x2y - 9y = 0
 y(6x-x^2-9)=0
y(x+3)(x+3)= 0

hence y is 0 or x is -3 or both

statement 1 is insufficient 
statement 2 just tells us x is negative which we already know so again insufficient 
together they are Sufficient hence C

ExpertsGlobal5 · Answer

It seems you have made a sign error in the factorization.
6x - x^2 - 9 = 0 becomes x^2 - 6x + 9 = 0, which factors as (x - 3)^2, not (x + 3)^2.
So, the solutions are y = 0 or x = 3, not -3.
With the correct factor, statement (2) is sufficient.

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

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GMAT Data Sufficiency (DS) Questions

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

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