已知x+y分之xy=1,y+z分之yz=2,z+x分之zx=3,求x+y+z的值

由已知得
(x+y)/(xy)=1
(y+z)/(yz)=1/2
(z+x)/(zx)=1/3
变形：
1/x+1/y=1 (1)
1/y+1/z=1/2 (2)
1/z+1/x=1/3 (3)
[(1)+(2)+(3)]/2
1/x+1/y+1/z=11/12 (4)
(4)-(2)
1/x=5/12 x=12/5
(4)-(3)
1/y=7/12 y=12/7
(4)-(1)
1/z=-1/12 z=-12
x+y+z=12/5+12/7-12=-276/35

xy=x+y,可得x=y/(y-1),代入zx=3(z+x),得2yz=3z-3y,将yz=2(y+z)代入2yz=3z-3y，可得z=-7y，代入yz=2(y+z)，可得y=12/7,x=12/5,z=-12

(XY)/(X+Y)=1 => (X+Y)/XY=1/X +1/Y=1
(YZ)/(Y+Z)=2=>(Y+Z)/YZ=1/Y +1/Z =1/2
(ZX)/(Z+X)=3=> (Z+X)/ZX=1/Z +1/X=1/3 相加 2(1/X +1/Y+ 1/Z)=11/6 =>1/X +1/Y+ 1/Z=11/12
=> 1/Z=-1/12 => Z=-12 ; 1/X=5/12 => X=12/5 ; 1/Y=7/12 =>Y=12/7
=>X+Y+Z=-276/35

因为XY)/(X+Y)=1 所以 (X+Y)/XY=1/X +1/Y=1
同理(YZ)/(Y+Z)=2,(Y+Z)/YZ=1/Y +1/Z =1/2
(ZX)/(Z+X)=3, (Z+X)/ZX=1/Z +1/X=1/3
三式相加 :2(1/X +1/Y+ 1/Z)=11/6 ,1/X +1/Y+ 1/Z=11/12,
1/Z=11/12-1 =-1/12, Z=-12 ;
同理1/X=5/12 , X=12/5 ; 1/Y=7/12 ,Y=12/7
所以X+Y+Z=-276/35