已知:(x+y)/z=(x+z)/y=(z+y)/x,且xyz不等于0,则分式(x+y)(x+z)(z+x)/xyz的值为?

来源：学生作业帮助网编辑：作业帮时间：2024/04/28 06:14:20

已知:(x+y)/z=(x+z)/y=(z+y)/x,且xyz不等于0,则分式(x+y)(x+z)(z+x)/xyz的值为?
已知:(x+y)/z=(x+z)/y=(z+y)/x,且xyz不等于0,则分式(x+y)(x+z)(z+x)/xyz的值为?

已知:(x+y)/z=(x+z)/y=(z+y)/x,且xyz不等于0,则分式(x+y)(x+z)(z+x)/xyz的值为?
(x+y)/z=(x+z)/y=(z+y)/x x,y,z等价 x=y=z
(x+y)(x+z)(z+x)/xyz=8

x=y=z
8

:(x+y)/z=(x+z)/y=(z+y)/x=（x+y+x+z+z+y)/(x+y+z)=2, 则原式等于8

设(x+y)/z=(x+z)/y=(y+z)/x=a,
则x+y=az,y+z=ax,x+z=ay,
三式相加得2(x+y+z)=a(x+y+z),
即(a-2)(x+y+z)=0,
所以a=2或x+y+z=0,
当a=2时有(x+y)(y+z)(x+z)/xyz=8xyz/xyz=8,
当x+y+z=0则x+y=-z,y+z=-x,x+z=-y,
所以(x+y)(y+z)(x+z)/xyz=-xyz/xyz=-1.

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