Problem: For all non-zero real numbers xxx and yyy such that x−y=xy,1x−1yx-y=x y, \dfrac{1}{x}-\dfrac{1}{y}x−y=xy,x1−y1 equals
Answer Choices:
A. 1xy\dfrac{1}{x y}xy1
B. 1x−y\dfrac{1}{x-y}x−y1
C. 000
D. −1-1−1
E. y−xy-xy−x Solution:
1x−1y=yxy−xxy=−xyxy=−1\dfrac{1}{x}-\dfrac{1}{y}=\dfrac{y}{x y}-\dfrac{x}{x y}=\dfrac{-x y}{x y}=-1x1−y1=xyy−xyx=xy−xy=−1.