Question

已知xy/x+y=3,yz/y+z=2,xz/x+z=1,求x，y，z的值

Answer 1

已知xy/(x+y)=3，yz/(y+z)=2，xz/(x+z)=1，求x，y，z的值
解：xy=3x+3y........(1)；yz=2(y+z)..........(2)；xz=x+z..........(3)
由(3)得z=x/(x-1)........(4)
将(4)代入(2)式得xy/(x-1)=2[y+x/(x-1)]=2[y(x-1)+x]/(x-1)，消去分母得xy=2(xy-y+x)；
化简得xy=-2x+2y.........(5)；
(1)-(5)得5x+y=0，故y=-5x；代入(5)得-5x²=-2x-10x，即有5x²-12x=x(5x-12)=0，
故x=12/5；或x=0(舍去)；
故x=12/5，y=-12；z=(12/5)/(12/5-1)=12/7.

Answer 2

xy/(x+y)=1,取倒数
(x+y)/xy=1
x/xy+y/xy=1
1/y+1/x=1.............1

yz/(y+z)=2,取倒数
(y+z)/yz=1/2
y/yz+z/yz=1/2
1/z+1/y=1/2...............2

xz/(x+z)=3取倒数
(x+z)/xz=1/3
x/xz+z/xz=1/3
1/z+1/x=1/3.................3

1式+2式+3式得
2(1/x+1/y+1/z)=1+1/2+1/3
2(1/x+1/y+1/z)=11/6
1/x+1/y+1/z=11/12...............4
将3式代入4式得
1/y+1/3=11/12
1/y=7/12
y=12/7

将2式代入4式得
1/x+1/2=11/12
1/x=5/12
x=12/5

Answer 3

将三个式子倒过来
(x+y)/xy=1/3
1/x +1/y=1/3 (4)
1/y+1/z=1/2 (5)
1/x+1/z=1 (6)
(4)+(5)+(6)得
2(1/x+1/y+1/z)=11/6
∴1/x+1/y+1/z=11/12 (7)
(7)-(5)得
1/x=5/12
把1/x=5/12代入(4)得
1/y=-1/12
把1/x=5/12代入(6）得
1/z=7/12
∴x=12/5
y=-12
z=12/7