已知x,y,z为实数,满足x+2y-z=6x-y+2z=3,那么x2+y2+z2的最小值是 _

问题描述:

已知x,y,z为实数,满足

x+2y-z=6
x-y+2z=3
,那么x2+y2+z2的最小值是 ______

x+2y-z=6①
x-y+2z=3②

①×2+②,得x+y=5,则y=5-x③,
①+2×②,得x+z=4,则z=4-x④,
把③④代入x2+y2+z2得,
x2+(5-x)2+(4-x)2
=3x2-18x+41
=3(x-3)2+14,
∴x2+y2+z2的最小值是14,
故答案为14.