If |x|=|y| and x=y which of the following must be true?
I. x+y=0
II. xy<0
III. x2=y2

Rationale

x^2 = y^2 must be true if |x| = |y| and x = y.

When both conditions |x| = |y| and x = y are satisfied, it directly implies that the squares of x and y are equal, leading to the conclusion that x^2 = y^2. This is because squaring both sides of x = y preserves equality.

A (III only) YOUR ANSWER

This choice suggests that only x^2 = y^2 is true, but it ignores the fact that conditions I and II are also satisfied when x = y. Therefore, while III is true, it is not the only valid statement under the given conditions.

B (I and II only) YOUR ANSWER

This choice claims that both x + y = 0 and xy < 0 must be true. However, if x = y, then x + y = 2x, which cannot equal 0 unless x itself is 0. Additionally, for x = y, the product xy is non-negative, contradicting the requirement that xy < 0. Thus, this option is incorrect.

C (I and III only) YOUR ANSWER

This choice implies that both x + y = 0 and x^2 = y^2 must be true. While x^2 = y^2 is indeed true, x + y = 0 is not necessarily valid unless x is specifically 0. Therefore, this choice is also incorrect.

D (II and III) CORRECT

This choice states that both xy < 0 and x^2 = y^2 must be true. However, when x = y, xy is actually non-negative, making this statement false as well. Thus, this option is misleading.

Conclusion

Given the conditions |x| = |y| and x = y, the only universally true statement is x^2 = y^2. Other statements, such as x + y = 0 and xy < 0, depend on the specific values of x and y and do not hold true in all cases. Therefore, the implication of equality solely leads to the conclusion about their squares.

If |x|=|y| and x=y which of the following must be true? I. x+y=0 II. xy<0 III. x2=y2

x^2 = y^2 must be true if |x| = |y| and x = y.

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If |x|=|y| and x=y which of the following must be true?
I. x+y=0
II. xy<0
III. x2=y2