3x-y z=0x 2y-z=6x y z=12
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x&3-y&3-z&3=3xyzx²=0(y+z),x=03(y+z)=0因为想,x,y,z正整数所以y=0z=0xy+yz+xz=(0)
令x/3=y/4=z/6=k≠0则x=3ky=4kz=6kxy+yz+xz=12k^2+18k^2+24k^2=54k^2x^2+y^2+z^2=(9+16+36)k^2=51k^2xy+yz+xz/
f(x,y,z)=yz+xz使得,y^2+z^2=1,yz=3令F(x,y,z)=yz+xz+a(y²+z²-1)+b(yz-3)Fx=z=0Fy=z+2ay+bz=0Fz=y+x
x^2-yz-8x+7=0……(1),y^2+z^2+yz-6x+6=0……(2);(1)×3+(2)得到:(y-z)^2=-3x^2+30x-27=-3(x-1)(x-9)>=0所以:1
4x-3y=3z.(1)x-3y=z.(2)(1)-(2)得3x=2zx=(2/3)z代入(2)得(2/3)z-3y=zy=-(1/9)z则xy+2yz/x²+y²+z²
X=1,Y=2,Z=3其实很简单!
xy\X+Y=12\71/y+1/x=7/12(1)YZ\Y+Z=6\51/z+1/y=5/6(2)XZ\X+Z=4\31/z+1/x=3/4(3)由(1)-(2)得1/x-1/z=-1/4(4)由(
xy/x+y=4,取倒数得:1/Y+1/X=1/4.[1]yz/y+z=6,取倒数得:1/Y+1/Z=1/6,.[2]zx/z+x=3,取倒数得:1/X+1/Z=1/3,.[3][1]-[2]:1/X
y=4/3XZ=2X带进去算就可以了
|x-2|+(y+3)²=0都是非负式所以分别都=0所以x-2=0y+3=0所以x=2y=-3又因为z是最大的负整数所以z=-1原式=2(x²y+xyz)-3(x²y-x
xy+yz+xz=4x+4y+6y+6z+3x+3zxy+yz+xz=7x+10y+9z
设k=x/3=y/4=z/6∴x=3k,y=4k,z=6k(xy+yz+xz)/(x²+y²+z²)=(12k²+24k²+18k²)/(9
令X=3k,由于x:y:z=3:4:6则:y=4k,z=6k将x=3k,y=4k,z=6k代入(xy+yz+xz)/(x^+y^+z^)则有:(xy+yz+xz)/(x^+y^+z^)=(12k^+2
由(x+y)/xy=5/6(y+z)/yz=-2/3得1/x+1/y=5/6①1/y+1/z=-2/3②1/x-1/z=3/21/z=1/x-3/2③z=2x/(2-3x)代入2(x+3)+xz=0化
-x=3,/y/=4,z+3=0,x=-3y=±4z=-3当y=4时xy+yz+xz=-12-12+9=-24+9=-15当y=-4时xy+yz+xz=12+12+9=24+9=33
图片中的题可以用琴森不等式构造函数f(x)=e^x/(3e^x+1)^0.5可以验证f``(x)>0对所有x成立因此f(x)是下凸函数有f(x)+f(y)+f(z)>=3f(x+y+z/3)令x=ln
(x+y+z)²=6²x²+y²+z²+2(xy+yz+zx)=36x²+y²+z²=36-22=14x³+y
(x+2y-z)^2+(z-x)^2=0所以x+2y-z=0,z-x=0x=z所以2y=0,y=0代入xz^2+yz-5√(xz^2+yz+9)+3=0x^3-5√(x^3+9)+3=0(x^3+9)
解题思路:本题的关键是将三个方程两边取倒数,化简后分别将方程等号左边和右边相加,得到1/x+1/y+1/z的值,最后将要求的分式化简,把1/x+1/y+1/z的值带入即可。解题过程:
|x-3|+|y+z|+|2z+1|=0则|x-3|=0x=3|y+z|=0y=-z=1/2|2z+1|=0z=-1/2xy-yz=3x1/2-1/2x(-1/2)=7/4