If $8x^3 + 27y^3 +64z^3 = 72 xyz$,  then the relation between x, y and z can be :

2x + 3y = -4z

Given that,

8x³ + 27y³ + 64z³ = 72 xyz,

a³+b³+c³−3abc=(a+b+c)(a²+b²+c²−ab−bc−ca)

and if a + b + c = 0 than,

a³+b³+c³=3abc

 (2x)³+(3y)³+(4z)³ = 3(2x)(3y)(4z) = 72xyz

So it is satisfying the condition a³+b³+c³=3abc

so, a = 2x , b = 3y ,c = 4z and  a + b + c = 0

2x + 3y + 4z = 0

We can write this also as =  2x + 3y = -4z