已知y+z/x=z+x/y=x+y/z,且x+y+z不等于0,求x+y-z/x+y+z的值.

Question

已知y+z/x=z+x/y=x+y/z,且x+y+z不等于0,求x+y-z/x+y+z的值.

Answer 1

设(y+z)/x=(z+x)/y=(x+y)/z=k
∵(y+z)/x=k
∴y+z=kx
同理，x+z=ky,x+y=kz
三式相加，得
(y+z)+(x+z)+(x+y)=kx+ky+kz
即 2(x+y+z)=k(x+y+z)
∵x+y+z≠0，两边约去x+y+z，得 k=2
∵x+y=kz
∴x+y=2z
∴(x+y-z)/(x+y+z)=(2z-z)/(2z+z)=z/3z=1/3