If `x , y , z` are not all zero such that `a x+b y+c z=0` `c x+a y+b z=0` `c x+a y+b z=0` then find `x^3:y^3:z^3`

Explanation

We have given:`a x+y+z=0`,
`x+b y+z=0`,
`x+y+c z=0`

Now add all the equations we get

`(a+b+c)x+(a+b+c)y+(a+b+c)z=0`
`(x+y+z)(a+b+c)=0x+y+z=0`
​`implies x=y=z` as `x^3+y^3+z^3-3xyz=(x+y+z).1/2[(x-y)^2+(y-z)^2+(z-x)^2]`

`implies x^3:y^3:z^3=1:1:1`