Problem:
If x−zy=zx+y=yx for three positive numbers x,y and z, all different, then yx=
Answer Choices:
A. 21
B. 53
C. 32
D. 35
E. 2
Solution:
A property of proportions is: If ba=dc=fe=k, then b+d+fa+c+e=k. Use this to obtain yx=(x−z)+z+yy+(x+y)+x=x+y2x+2y=2.
Note. In general, x=4t,y=2t and z=3t for t>0.
OR
Simplify x−zy=yx and zx+y=yx to obtain y2=x2−xz and xy+y2= xz, respectively. Substitute for xz to find that y2=x2−(xy+y2), or 0=x2−xy−2y2=(x−2y)(x+y). Thus yx=2 or yx=−1. Since x,y>0, it follows that yx=2.