Bunuel wrote:

If x=−|y|, which of the following must be true?

A. y=−x

B. x=y

C. |x|=−y

D. x^2=y^2

E. x^3=y^3

We are given that x = -|y|, and must determine which answer choice MUST BE TRUE. We can use numerical values for x and y so long they satisfy the equation x = -|y|, and then use them to test each answer choice.

A) y = −xAnswer choice A does not have to be true. For instance, if x = -2 and y = -2, then -2 ≠ 2 and answer choice A is not true.

B) x = yAnswer choice B does not have to be true. For instance, if x = -2 and y = 2, then -2 ≠ 2 and answer choice B is not true.

C) |x| = −yAnswer choice C does not have to be true. For instance, if x = -2 and y = 2, then |-2| ≠ −2 and answer choice C is not true.

D) x^2=y^2Answer choice D MUST BE TRUE. Using some of the examples for x and y above, we see that in all cases answer choice D must be true.

If x = -2 and y = -2, then (-2)^2 = (-2)^2, and if x = -2 and y = 2, then (-2)^2 = (2)^2.

In fact, no matter what values for x and y we use (which satisfy the equation x = -|y|), x^2 WILL ALWAYS EQUAL y^2.

Since answer choice D must be true, we can stop here.

Answer: D

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