已知xyz=1,求x/(xy+x+1)+y/(yz+y+1)+z/(xz+z+1)的值

题目

已知xyz=1,求x/(xy+x+1)+y/(yz+y+1)+z/(xz+z+1)的值
要理由

答案

xyz=1
所以
z=1/xy
xz=1/y
yz=1/x
x/(xy+x+1)+y/(yz+y+1)+z/(xz+z+1)
=x/(xy+x+1)+y/(1/x+y+1)+(1/xy)/(1/y+1/xy+1)
第二个分子分母同乘以x
第三个分子分母同乘以xy
=x/(xy+x+1)+xy/(xy+x+1)+1/(xy+x+1)
=(xy+x+1)/(xy+x+1)
=1