Bunuel
If x·y≠0, what is the value of x/y?
(1) −|x|=y
(2) −|y|=x
If x·y≠0, what is the value of x/y?xy≠0 implies that neither of them is 0.
(1) −|x|=y. Notice that this implies that y must be a negative number.
If \(x>0\), then \(-x=y\) --> \(\frac{x}{y}=-1\), for example, consider \(x=-1\) and \(y=-1\).
If \(x<0\), then \(-(-x)=y\) --> \(\frac{x}{y}=1\), for example, consider \(x=1\) and \(y=-1\).
Not sufficient.
(2) −|y|=x. Notice that this implies that x must be a negative number.
If \(y>0\), then \(-y=x\) --> \(\frac{x}{y}=-1\), for example, consider \(x=-1\) and \(y=1\).
If \(y<0\), then \(-(-y)=x\) --> \(\frac{x}{y}=1\), for example, consider \(x=-1\) and \(y=-1\).
Not sufficient.
(1)+(2) Since from (2) we have that x is a negative number, then from (1) -|x|=y, becomes -(-x)=y --> x=y --> x/y=1. Sufficient.
Answer: C.
Hope it's clear.