第一題x+2/y=y+2/z=√2 求xyz=?
x=√2-2/y (1)
y=√2-2/z (2)
(2)代入(1)
x=√2 -2/(√2-2/z)=√2-2z/(√2z-2)=(2z-2√2-2z)/(√2z-2)
=-2√2/(√2z-2) (3)
xyz=-2√2/(√2z-2)*(√2-2/z)z=-2√2/(√2z-2)*(√2z-2)=-2√2
第二題x+y+z=3 1/x+1/y+1/z=1/3 求(x-3)(y-3)(z-3)=?x³+y³+z³=?
x+y+z=3
1/x+1/y+1/z=(yz+xz+xy)/xyz=1/3
3xy+3yz+3xz=xyz
(x-3)(y-3)(z-3)=(xy-3x-3y+9)(z-3)=xyz-3xy-3xz+9x-3yz+9y+9z-27
=xyz-(3xy+3yz+3xz)+9(x+y+z)-27=xyz-xyz+9*3-27=0
(x+y+z)^2=(x+y+z)(x+y+z)=x^2+xy+xz+xy+y^2+yz+xz+yz+z^2
=x^2+y^2+z^2+2xy+2yz+2xz
(x+y+z)^3=(x+y+z)^2*(x+y+z)=(x^2+y^2+z^2+2xy+2yz+2xz)(x+y+z)
=x^3+yx^2+zx^2+xy^2+y^3+zy^2+xz^2+yz^2+z^3+2yx^2+2xy^2+2xyz+2xyz+2zy^2
+2yz^2+2zx^2+2xyz+2xz^2
=x^3+y^3+z^3+3xy^2+3yz^2+3xz^2+3yx^2+3zy^2+3zx^2+3xyz+3xyz
=x^3+y^3+z^3+3xy(y+x+z)+3yz(z+y+x)+3xz(z+x+y-y)
=x^3+y^3+z^3+9xy+9yz+3xz(3-y)
=x^3+y^3+z^3+3(3xy+3yz+3xz)-3xyz
=x^3+y^3+z^3+3xyz-3xyz=x^3+y^3+z^3
x^3+y^3+z^3=(x+y+z)^3=3^3=27