Official Guide Explanation:
Problem Solving #106




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Solution and Metadata

Question: 106
Page: 75
Difficulty: 5 (Moderate)
Category 1: Arithmetic > Real Numbers >
Category 2: Algebra > Equations >

Explanation: Trying to simplify the equation x2 = xy raises an important point. Generally, you can divide both sides by x, leaving x = y. However, this question specifies that x and y are different integers, so x ≠ y. Whenever you divide both sides of an equation by a variable, there's always the possibility that the variable is 0. (If both sides are multiplied by zero, both sides are equal to zero, and they are equal.) In this case, if x = 0, y could be any number. So, if x2 = xy and x ≠ y, it must be the case that x = 0. Now, we can look at the statements:

I.    Yes, we've established that x = 0.

II.    This cannot be true: if x = 0 and x ≠ y, y cannot be 0.

III.    Since x = 0, if x= - y, then y = 0, and we've just seen that y cannot be 0.

I is the only statement that must be true, so the correct choice is (A).

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