I have a somewhat seemingly simple set of nonlinear equations that need to be solved given by
$$ \begin{array}{ccccccc} x & + & m_1 y & + & m_2 z & = & 0\\ \frac{x}{m_1} & + & y & + & m_3 z & = & 0\\ \frac{x}{m_2} & + & \frac{y}{m_3} & + & z & = & 0 \end{array} $$
where $(x, y, z)$ are known constants and $(m_1, m_2, m_3)$ are the values that need to be solved for. Is there any chance there is an analytical way to solve these regardless of the values of $(x, y, z)$? If not, how do I reduce this down to an equation/form that I might be able to solve it numerically?
This expands on @Rodrigo de Azevedo's comment. Clearing the denominators, we get a system of algebraic equations (+ a condition $m_1 m_2 m_3 \not = 0$): $\begin{cases}m_1y+m_2z+x=0\\ m_1y+m_1m_3z+x=0 \\ m_3x+m_2y+m_2m_3z=0\end{cases}$
Computations with systems of algebraic equations are usually done nowadays via Groebner bases, but for this example the general theory would be a sledgehammer.
As @Ninad Munshi notes, the first two equations imply that either $z = 0$ or $m_1 m_3 = m_2$. If $z = 0$, $\begin{cases}m_1y+x=0\\ m_3x+m_2y=0\end{cases}$ is a system of linear equations (mind the possibly zero coefficients $x,y$!) If $m_1 m_3 = m_2$, by eliminating $m_2$ we get a single equation $m_1y+m_1m_3z+x=0$.