How to solve this set of equations?

[emielzij]

Hello,

I'm in need to solve the following set:

a/x² + b/y² + c/z² = 0
d/x³ + e/y³ + f/z³ = 0

The coëfficiënts a,b,c,d,e,f are real and I'm only intrested in real solutions for x,y,z.

How can I find the combination of x y and z such that both of the equations are simultaneously satisfied given the coëfficiënts?

 

[ksdhart2]

Well, there's the trivial solution \(a = b = c = d = e = f = 0\) which works for any choice of \(x, y, z\) (so long as none of them are 0). Otherwise, there's not much that can be done with nine variables and only two equations. The best you can come up with is to arbitrarily choose any two of them and them solve for the other seven in terms of those two chosen variables.

 

[Dr.Peterson]

As I read it, there are three variables and two equations, so the best you could do is to solve for, say, x and y in terms of z. Solve one equation for z, put that into the other equation, and then hope the resulting equation can be solved.

Do you have reason to think more could be done? If this is part of a larger problem, it may be necessary to see the whole problem in order to make a good suggestion.