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x, y and z are real numbers with xyz = 1. What is the value of (x - 1)(y - 1)(z - 1)?

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by Max@Math Revolution » Mon Mar 30, 2020 12:17 am

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[GMAT math practice question]

x, y and z are real numbers with xyz = 1. What is the value of (x - 1)(y - 1)(z - 1)?

1) x + y + z = 3
2) xy + yz + zx = -4
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Source: — Data Sufficiency |
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Re: x, y and z are real numbers with xyz = 1. What is the value of (x - 1)(y - 1)(z - 1)?

by Max@Math Revolution » Wed Apr 01, 2020 1:02 am
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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

(x - 1)(y - 1)(z - 1)
= (x – 1)(yz – y – z + 1)
= xyz – xy – xz + x – yz + y + z - 1
= xyz – (xy + yz + zx) + (x + y + z) – 1

Since we have x + y + z = 3 and xy + yz + zx = -4 from both conditions 1) and 2), we have (x - 1)(y - 1)(z - 1) = xyz – (xy + yz + zx) + (x + y + z) – 1 = 1 – (-4) + 3 – 1 = 7.
Since both conditions together yield a unique solution, they are sufficient.

Therefore, C is the answer.
Answer: C
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