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(x – 1)(y – 1) = 1?

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(x – 1)(y – 1) = 1?

by PGMAT » Sat Apr 14, 2012 3:59 pm

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If xy > 0, does (x - 1)(y - 1) = 1?
(1) x + y = xy
(2) x = y

OA A

Can some one explain why statement 2 is not sufficient? Since xy > 0, we can arrive at a value of x=y=2 and therefore, (x-1)(y-1) = 1, which means its sufficient. But the OA says otherwise.

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by sam2304 » Sat Apr 14, 2012 7:15 pm

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PGMAT wrote:If xy > 0, does (x - 1)(y - 1) = 1?
(1) x + y = xy
(2) x = y

OA A

Can some one explain why statement 2 is not sufficient? Since xy > 0, we can arrive at a value of x=y=2 and therefore, (x-1)(y-1) = 1, which means its sufficient. But the OA says otherwise.

Thanks.
I can say x = y = 3 and (x-1)(y-1) = 4, which is INSUFFICIENT.

Hope you get where you go wrong. You are considering only the choice where the eqn gets satisfied but you have innumerable options. We should get a definite value or Yes/No, in this case its either yes or no. With your choice of nos we get a YES and with mine we get a NO, so it is INSUFFICIENT.
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by Shalabh's Quants » Sat Apr 14, 2012 9:15 pm

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PGMAT wrote:If xy > 0, does (x - 1)(y - 1) = 1?
(1) x + y = xy
(2) x = y

OA A

Can some one explain why statement 2 is not sufficient? Since xy > 0, we can arrive at a value of x=y=2 and therefore, (x-1)(y-1) = 1, which means its sufficient. But the OA says otherwise.

Thanks.
By Stat. 2...

Put x = y in (x - 1)(y - 1) = 1, we get x^2-1 = 1

or x^2 = 2. Still we land up in situation of Is x^2 = 2 ?. Not sufficient.
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by Birottam Dutta » Sat Apr 14, 2012 9:29 pm

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Exactly!

The learning from this problem should be that you may have a million cases to prove an equation correct but you only need one to falsify it!

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by Anurag@Gurome » Sun Apr 15, 2012 7:59 pm

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PGMAT wrote:If xy > 0, does (x - 1)(y - 1) = 1?
(1) x + y = xy
(2) x = y

OA A

Can some one explain why statement 2 is not sufficient? Since xy > 0, we can arrive at a value of x=y=2 and therefore, (x-1)(y-1) = 1, which means its sufficient. But the OA says otherwise.

Thanks.
Question is: Is (x - 1)(y - 1) = 1? or Is xy - x - y + 1 = 1 or Is xy - x - y = 0 or Is xy = x + y?

(1) x + y = xy directly answers the question: SUFFICIENT.

(2) x = y
So, the question becomes is x² = 2x or is x(x - 2) = 0 or Is x = 0 or x = 2?
It is given that xy > 0, so x cannot be equal to 0.
Now the question becomes: Is x = 2?, bit we have NOT sufficient info to answer this question.

The correct answer is A.
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by Brent@GMATPrepNow » Wed Aug 15, 2018 5:41 am

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PGMAT wrote:If xy > 0, does (x - 1)(y - 1) = 1?

(1) x + y = xy
(2) x = y
Given: xy > 0

Target question: Does (x - 1)(y - 1) = 1?

This is a good candidate for rephrasing the target question
Take the equation: (x - 1)(y - 1) = 1
Use FOIL to expand the left side to get: xy - x - y + 1 = 1
Subtract 1 from both sides to get: xy - x - y = 0

REPHRASED target question: Does xy - x - y = 0?

Statement 1: x + y = xy
Take the REPHRASED target question and replace xy with x+y to get: Does (x + y) - x - y = 0?
Simplify the left side to get: Does 0 = 0?
Great! The answer to the REPHRASED target question is YES, it IS the case that xy - x - y = 0
Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT

Statement 2: x = y
There are several values of x and y that satisfy statement 2. Here are two:
Case a: x = 2 and y = 2. In this case, the equation xy - x - y = 0 becomes (2)(2) - 2 - 2 = 0, which works!
So, the answer to the REPHRASED target question is YES, it IS the case that xy - x - y = 0

Case b: x = 1 and y = 1. In this case, the equation xy - x - y = 0 becomes (1)(1) - 1 - 1 = 0, which does NOT work.
So, the answer to the REPHRASED target question is NO, it is NOT the case that xy - x - y = 0

Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

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by GMATGuruNY » Wed Aug 15, 2018 9:06 am

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PGMAT wrote:If xy > 0, does (x - 1)(y - 1) = 1?
(1) x + y = xy
(2) x = y
Alternate approach:

Statement 1:
Case 1: x=2
Substituting x=2 into x+y = xy, we get:
2 + y = 2y
2 = y.
In this case, (x-1)(y-1) = (2-1)(2-1) = 1, so the answer to the question stem is YES.

Case 2: x=3
Substituting x=3 into x+y = xy, we get:
3+ y = 3y
3 = 2y
3/2 = y.
In this case, (x-1)(y-1) = (3-1)(3/2 - 1) = (2)(1/2) = 1, so the answer to the question stem is YES.

Case 3: x=-2
Substituting x=-2 into x+y = xy, we get:
-2 + y = -2y
3y = 2
y = 2/3.
In this case, (x-1)(y-1) = (-2-1)(2/3 - 1) = (-3)(-1/3) = 1, so the answer to the question stem is YES.

In every case, the answer to the question stem is YES, implying that Statement 1 is SUFFICIENT.

Statement 2:
Case 1 also satisfies Statement 2.
In Case 1, the answer to the question stem is YES.

Case 4: x=y=1
In this case, (x-1)(y-1) = 0, so the answer to the question stem is NO.

Since the answer is YES in Case 1 but NO in Case 4, INSUFFICIENT.

The correct answer is A.
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