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