已知p＋q＋r＝9,且p/(x^2-yz)=q/(y^2-zx)=r/(z^2-xy),则(px+qy+rz)/(x+y+z)等于?为什么能列出p/(x^2-yz)=q/(y^2-zx)=r/(z^2-xy)=(px+qy+rz)/(x^3+y^3+z^3-3xyz)?过程具体点.

Question

已知p＋q＋r＝9,且p/(x^2-yz)=q/(y^2-zx)=r/(z^2-xy) ,则(px+qy+rz)/(x+y+z)等于?
为什么能列出p/(x^2-yz)=q/(y^2-zx)=r/(z^2-xy)=(px+qy+rz)/(x^3+y^3+z^3-3xyz)?过程具体点.

Answer 1

p/(x^2-yz)=px/(x^3-xyz)分子、分母同乘x
q/(y^2-zx)=qy/(y^3-yzx)分子、分母同乘y
r/(z^2-xy)=rz/(z^3-zxy)分子、分母同乘z
(a/b=c/d=(a+c)/(b+d)比例性质)
所以
p/(x^2-yz)=q/(y^2-zx)=r/(z^2-xy)=(px+qy+rz)/(x^3+y^3+z^3-3xyz)
p/(x^2-yz)=(px+qy+rz)/(x^3+y^3+z^3-3xyz)
q/(y^2-zx)=(px+qy+rz)/(x^3+y^3+z^3-3xyz)
r/(z^2-xy)=(px+qy+rz)/(x^3+y^3+z^3-3xyz)
p=(x^2-yz)(px+qy+rz)/(x^3+y^3+z^3-3xyz)
q=(y^2-zx)(px+qy+rz)/(x^3+y^3+z^3-3xyz)
r=(z^2-xy)(px+qy+rz)/(x^3+y^3+z^3-3xyz)
p+q+r=(px+qy+rz)(x^2+y^2+z^2-xy-xz-yz)/x^3+y^3+z^3-3xyz=9
(x+y+z)(x^2+y^2+z^2-xy-xz-yz)=x^3+y^3+z^3-3xyz
(px+qy+rz)/(x+y+z)=9