Problem: If x and y are non-zero numbers such that x=1+y1​ and y=1+x1​, then y equals
Answer Choices:
A. x−1
B. 1−x
C. 1+x
D. −x
E. x
Solution:
Method I. By subtraction we obtain x−y=y1​−x1​=xyx−y​ ∴(x−y)(1−xy1​)=0,x=y (The result y=x1​ is rejected. Why?)
Method II. xy=y+1 and xy=x+1∴y+1=x+1,x=y.
Method III. Since y=1+x1​ and x=1+y1​,y=1+1+y1​1​ and x=1+1+x1​1​.
Therefore, y2−y−1=0 and x2−x−1=0. Let the roots of z2−z−1=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.∴y=x.
Note. For all three methods, since y=xî€ =0,y cannot equal any of the other choices shown.