若x+y+z=0，求x（1y+1z）+y（1x+1z）+z（1x+1y）的值．

Question

若x+y+z=0，求x（(1)/(y)+(1)/(z)）+y（(1)/(x)+(1)/(z)）+z（(1)/(x)+(1)/(y)）的值．

Answer 1

原式=(x)/(y)+(x)/(z)+(y)/(x)+(y)/(z)+(z)/(x)+(z)/(y)
=(x+z)/(y)+(x+y)/(z)+(y+z)/(x)
∵x+y+z=0，
∴x+z=-y，x+y=-z，y+z=-x，
∴原式=(−y)/(y)+(−z)/(z)+(−x)/(x)
=-1-1-1
=-3．