已知三个数x,y,z满足xy/x+y=-2,yz/y+z=4/3,zx/z+x=-4/3.求xyz/xy+yz+zx的值.

问题描述:

已知三个数x,y,z满足

xy
x+y
=-2,
yz
y+z
=
4
3
zx
z+x
=-
4
3
.求
xyz
xy+yz+zx
的值.

xy
x+y
=-2,
yz
y+z
=
4
3
zx
z+x
=-
4
3

1
x
+
1
y
=-
1
2
1
y
+
1
z
=
3
4
1
z
+
1
x
=-
3
4

∴2(
1
x
+
1
y
+
1
z
)=-
1
2
,即
1
x
+
1
y
+
1
z
=-
1
4

xyz
xy+yz+zx
=
1
1
x
+
1
y
+
1
z
=-4.