bubbliiiiiiii wrote:If x and y are integers and 4xy = x²y + 4y, what is the value of xy?

(1) y − x = 2
(2) x³ < 0

Rule:
If an equation includes an unknown that could be equal to 0, DO NOT DIVIDE BY THE UNKNOWN.

Here, it is possible that y=0, so we cannot divide 4xy = x²y + 4y by y.
Instead, we should COLLECT LIKE TERMS, as follows:
0 = x²y - 4xy + 4y
0 = y(x² - 4y + 4)
0 = y(x-2)².
The right side will be equal to 0 if y=0 or if x=2.
Question stem, rephrased:
If it is required that x=2 or that y=0, what is the value of xy?

Statement 1: y − x = 2
If x=2, then y=4.
In this case, xy = 2*4 = 8.
If y=0, then x=-2.
In this case, xy = (-2)(0) = 0.
Since xy can be different values, INSUFFICIENT.

Statement 2: x³ < 0
Since x=2 is not possible, the condition in red requires that y=0.
Thus, xy = (x)(0) = 0.
SUFFICIENT.

The correct answer is B.

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If x and y are integers and 4xy = x2y + 4y, what is the value of xy?

(1) y - x = 2
(2) x3 < 0

==> using the original condition we can get (x^2)y-4xy+4y=0, y(x^2-4x+4)=0, y(x-2)^2=0 therefore y=0 or x=2. Thus we have two variables (x,y) and one equation (4xy = x2y + 4y) meaning we would need one more equation, therefore D is likely the answer. .

When we actually solve the problem,

in case of 1), x=2, y=4 or y=0, x=-2 makes it not unique, thus is not sufficient
in case of 2, x^3<0==> x<0 makes it impossible for x to have value 2, thus y = 0 and xy=0. This is sufficient
Therefore the answer is B.



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Hi Mitch,
didn't understand how u r getting statement 2 as sufficient.

sandipgumtya wrote:Hi Mitch,
didn't understand how u r getting statement 2 as sufficient.

As shown in my post above, the equation in the question prompt (4xy = x²y + 4y) requires that x=2 or that y=0.

Statement 2: x³ < 0
Here, it is not possible that x=2.
Since the equation in the question prompt requires that x=2 or that y=0, and Statement 2 indicates that x≠2, it must be true that y=0.
Since y=0, xy=0.
Since the value of xy can be determined, Statement 2 is SUFFICIENT.