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If x and y are integers such that (xy)^2 + x^2 – 2xy – 2x + 2 = 0, wha

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If x and y are integers such that (xy)^2 + x^2 – 2xy – 2x + 2 = 0, wha  [#permalink]

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New post 25 Jul 2018, 01:12
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A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

65% (02:08) correct 35% (02:48) wrong based on 37 sessions

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[Math Revolution GMAT math practice question]

If \(x\) and \(y\) are integers such that \((xy)^2 + x^2 – 2xy – 2x + 2 = 0\), what is the value of \(y\)?

A. \(-2\)
B. \(-1\)
C. \(0\)
D. \(1\)
E. \(2\)

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If x and y are integers such that (xy)^2 + x^2 – 2xy – 2x + 2 = 0, wha  [#permalink]

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New post 25 Jul 2018, 01:23
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\((xy)^2 + x^2 – 2xy – 2x + 2 = 0\)

=> \((xy)^2 + x^2 – 2xy – 2x + 1 + 1 = 0\)

=> \((xy)^2 – 2(xy) + 1 + x^2 – 2x + 1 = 0\)

=> \((xy-1)^2 + (x-1)^2 = 0\)

=> (xy - 1) = 0 and (x - 1) = 0

=> y = \(\frac{1}{x}\) and x = 1

=> y = 1 and x = 1

Hence option D
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Re: If x and y are integers such that (xy)^2 + x^2 – 2xy – 2x + 2 = 0, wha  [#permalink]

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New post 25 Jul 2018, 03:41
MathRevolution wrote:
[Math Revolution GMAT math practice question]

If \(x\) and \(y\) are integers such that \((xy)^2 + x^2 – 2xy – 2x + 2 = 0\), what is the value of \(y\)?

A. \(-2\)
B. \(-1\)
C. \(0\)
D. \(1\)
E. \(2\)


We can PLUG IN THE ANSWERS, which represent the value of y.

B: y=-1
Substituting y=-1 into \((xy)^2 + x^2 – 2xy – 2x + 2 = 0\), we get:
\(x^2 + x^2 + 2x – 2x + 2 = 0\)
\(2x^2 + 2 = 0\)
\(x^2 + 1 = 0\)
\(x^2 = -1\)
Since the square of a value cannot be negative, eliminate B.

D: y=1
Substituting y=1 into \((xy)^2 + x^2 – 2xy – 2x + 2 = 0\), we get:
\(x^2 + x^2 - 2x – 2x + 2 = 0\)
\(2x^2 - 4x + 2 = 0\)
\(x^2 - 2x + 1 = 0\)
\((x-1)^2 = 0\)
\(x=1\)
Success!


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Re: If x and y are integers such that (xy)^2 + x^2 – 2xy – 2x + 2 = 0, wha  [#permalink]

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New post 27 Jul 2018, 00:52
=>

\((xy)^2 + x^2 – 2xy – 2x + 2 = 0\)
\(=> (xy)^2 – 2xy + 1 + x^2 – 2x + 1 = 0\)
\(=> (xy – 1)^2 + (x – 1)^2 = 0\)
\(=> xy = 1\) and \(x = 1\).
Thus , \(x = 1\) and \(y = 1\).

Therefore, the answer is D.
Answer: D
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Re: If x and y are integers such that (xy)^2 + x^2 – 2xy – 2x + 2 = 0, wha &nbs [#permalink] 27 Jul 2018, 00:52
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