已知非零实数x,y,z满足x+y+z=0,求x(1/y+1/z)+y(1/x+1/z)+z(1/x+1/y)的值.谢.
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![已知非零实数x,y,z满足x+y+z=0,求x(1/y+1/z)+y(1/x+1/z)+z(1/x+1/y)的值.谢.](/uploads/image/z/12686943-39-3.jpg?t=%E5%B7%B2%E7%9F%A5%E9%9D%9E%E9%9B%B6%E5%AE%9E%E6%95%B0x%2Cy%2Cz%E6%BB%A1%E8%B6%B3x%2By%2Bz%3D0%2C%E6%B1%82x%281%2Fy%2B1%2Fz%29%2By%281%2Fx%2B1%2Fz%29%2Bz%281%2Fx%2B1%2Fy%29%E7%9A%84%E5%80%BC.%E8%B0%A2.)
已知非零实数x,y,z满足x+y+z=0,求x(1/y+1/z)+y(1/x+1/z)+z(1/x+1/y)的值.谢.
已知非零实数x,y,z满足x+y+z=0,求x(1/y+1/z)+y(1/x+1/z)+z(1/x+1/y)的值.谢.
已知非零实数x,y,z满足x+y+z=0,求x(1/y+1/z)+y(1/x+1/z)+z(1/x+1/y)的值.谢.
x(1/y+1/z)+y(1/x+1/z)+z(1/x+1/y)
=(y+z)/x+(x+z)/y+(x+y)/z
=(-x)/x+(-y)/y+(-z)/z
=-1-1-1
=-3
x+y+z=0 (x+y=-z x+z=-y y+z=-x)
x(1/y+1/z)+y(1/x+1/z)+z(1/x+1/y)
=x/y+x/z+y/x+y/z+z/x+z/y
=y/x+z/x+x/y+z/y+x/z+y/z
=(y+z)/x+(x+z)/y+(x+y)/z
=(-x/x)+(-y/y)+(-z/z)
=-1-1-1
=-3
x(1/y+1/z)+y(1/x+1/z)+z(1/x+1/y)乘以xyz得
x(xz+xy)+y(yz+xy)+z(yz+xz)
=(-y-z)(xz+xy)+y(yz+xy)+z(yz+xz)
=-xyz-xy²-xz²-xyz+y²z+xy²+yz²+xz²
=-2xyz+y²z+yz...
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x(1/y+1/z)+y(1/x+1/z)+z(1/x+1/y)乘以xyz得
x(xz+xy)+y(yz+xy)+z(yz+xz)
=(-y-z)(xz+xy)+y(yz+xy)+z(yz+xz)
=-xyz-xy²-xz²-xyz+y²z+xy²+yz²+xz²
=-2xyz+y²z+yz²
=yz(-2x+y+z)
=yz(-2x-x)
=-3xyz
所以
x(1/y+1/z)+y(1/x+1/z)+z(1/x+1/y)
=-3xyz/xyz
=-3
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