Question

已知,x,y,z为非零实数,且满足x+y-z/z=y+z-x/x=z+x-y/y,求x+y+z/z的值

Answer 1

解答：
设x+y-z/z=y+z-x/x=z+x-y/y=k
则：x+y-z=kz，y+z-x=kx，z+x-y=ky
三式相加，得到：x+y+z=k(x+y+z)，解得：k=1
所以：x+yy-z=z，解得：x+y=2z
所以：(x+y+z)/z=(2z+z)/z=3

Answer 2

x,y,z≠0
(x+y-z) / z = (y+z-x) / x = (z+x-y) / y
令x+y+z=m
(m-2z)/z = (m-2x)/x = (m-2y)/y
m/z - 2 = m/x - 2 = m/y - 2
m/z = m/x = m/y
x=y=z
(x+y+z)/z = 3z/z = 3

Answer 3

0或者3
X+Y+Z=0时 x+y+z/z=0
x+y+z不等于0时 x+y-z/z=y+z-x/x=z+x-y/y,根据等比性质 x+y+z/ x+y+z=1=x+y-z/z=y+z-x/x=z+x-y/y
x+y=2z x+y+z/z=3z/z=3