Problem: If x,yx, yx,y and y−1xy-\dfrac{1}{x}y−x1 are not 000, then
x−1yy−1x equals \dfrac{x-\dfrac{1}{y}}{y-\dfrac{1}{x}} \quad \text { equals } y−x1x−y1 equals
Answer Choices:
A. 111
B. xy\dfrac{x}{y}yx
C. yx\dfrac{y}{x}xy
D. xy−yx\dfrac{x}{y}-\dfrac{y}{x}yx−xy
E. xy−1xyxy-\dfrac{1}{xy}xy−xy1
Solution:
x−1yy−1x=xy−1yxy−1x=xy−1y⋅xxy−1=xy\dfrac{x-\dfrac{1}{y}}{y-\dfrac{1}{x}}=\dfrac{\dfrac{x y-1}{y}}{\dfrac{x y-1}{x}}=\dfrac{x y-1}{y} \cdot \dfrac{x}{x y-1}=\dfrac{x}{y}y−x1x−y1=xxy−1yxy−1=yxy−1⋅xy−1x=yx.