已知(x+y)(x+z)=x,(y+z)(y+x)=2y,(z+x)(z+y)=3z,求x,y,z

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已知(x+y)(x+z)=x,(y+z)(y+x)=2y,(z+x)(z+y)=3z,求x,y,z
已知(x+y)(x+z)=x,(y+z)(y+x)=2y,(z+x)(z+y)=3z,求x,y,z

已知(x+y)(x+z)=x,(y+z)(y+x)=2y,(z+x)(z+y)=3z,求x,y,z
显然
x=y=z=0是一解.
若x,y,z中有两数互为相反数,设为x,y(因为下面的过程忽略系数的影响,是对称的)
则由1,2两式得到x=y=0,则z=3.
于是又得到3组解(x,y,z)=(0,0,3),(1,0,0),(0,2,0)
若x,y,z,x+y,y+z,z+x都不为0,原方程的三个式子两两相除,可以解得x=5y=-7z
设x=35k,y=7k,z=-5k,代入原方程就知道k没有非0解.
故(x,y,z)=(0,0,3),(1,0,0),(0,2,0),(0,0,0)

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