x+y-z/z=x-y+z/y=y+z-x/x,且xyz≠0,求代数式(x+y)(y+z)(x+z)/xyz

Question

x+y-z/z=x-y+z/y=y+z-x/x,且xyz≠0,求代数式(x+y)(y+z)(x+z)/xyz

Answer 1

设x+y-z/z=x-y+z/y=y+z-x/x=k 有x+y-z=kz x-y+z=ky y+z-x=kx 三式相加得x+y+z=k（x+y+z） k=1 得x+y=（k+1）z x+z=（k+1）y y+z=（k+1）x (x+y)(y+z)(x+z)/xyz=（k+1）^3xyz/xyz=(k+1)^3=8