solving some complex equations I want to solve the following three equations for $x,y,z$:
$$\begin{eqnarray}k_1&=&\frac{x+y+z}{yz}\\
k_2&=&\frac{x^2+y^2+z^2}{y^2z^2}\\
k_3&=&\frac{x^3+y^3+z^3}{y^3z^3}\end{eqnarray}$$
where $k_1,k_2,k_3$ are constants.
Is there any kind of standard methods?

If you set $\frac{x}{yz} = p$, $\frac{1}{y} = q$, $\frac{1}{z} = r$, then your equations are
$p + q + r = k_1$
$p^2 + q^2 + r^2 = k_2$
$p^3 + q^3 + r^3 = k_3$
Using $(p+q+r)^2 - (p^2 + q^2 + r^2) = 2(pq + qr + rp)$ we get
$pq + qr + rp = (k_1^2 - k_2)/2$
Similarly we can find the value of $pqr$.
Thus we can find the polynomial $(x-p)(x-q)(x-r)$, by finding its coefficients. Once we have the polynomial, since it is a cubic, it is solvable by standard methods.
Once we find $p,q,r$, finding $x,y,z$ should not be difficult (keep in mind the permutations possible).
See: Newton's identities for a way to find the coefficients.
See: Roots of cubic for a formula for the roots of cubic.
