Problem: If x and y are non-zero numbers such that x=1+y1β and y=1+x1β, then y equals
Answer Choices:
A. xβ1
B. 1βx
C. 1+x
D. βx
E. x
Solution:
Method I. By subtraction we obtain xβy=y1ββx1β=xyxβyβ β΄(xβy)(1βxy1β)=0,x=y (The result y=x1β is rejected. Why?)
Method II. xy=y+1 and xy=x+1β΄y+1=x+1,x=y.
Method III. Since y=1+x1β and x=1+y1β,y=1+1+y1β1β and x=1+1+x1β1β.
Therefore, y2βyβ1=0 and x2βxβ1=0. Let the roots of z2βzβ1=0 be r and s. Then the given equations imply that, when x=r, so does y=r or that, when x=s, so does y=s.β΄y=x.
Note. For all three methods, since y=xξ =0,y cannot equal any of the other choices shown.