已知三个数x，y，z满足xyx+y=-2，yzy+z=43，zxz+x=-43．求xyzxy+yz+zx的值．

Question

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．

Answer 1

∵(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)，
则(xyz)/(xy+yz+zx)=(1)/((1)/(x)+(1)/(y)+(1)/(z))=-4．