Given two equations $$mn=r\tag{1}$$ $$rn=b\tag{2}$$ where $m$, $r$, and $b$ are polynomials of multiple variables, is it generally possible to express $n$ as a polynomial of the same variables as the other functions? To be clear, I mean an $n$ that can be expressed as a ratio of polynomials, where the denominator of this ratio is constant.
Bivariate example: If $r=xy$, $m=x$, and $b=y$, our equations become: $$xn=xy$$ $$xyn=y$$ Considering the latter result first, we find that $n=1/x$ for $x \ne 0$, which does not satisfy our polynomial requirement. However, considering the first of these two equations, we find that $n=y$, which satisfies our requirements. This is an illustration of how the multivariate case differs from the univariate one.
My work so far: Let's assume that the equation we use to define $n$ will be a linear combination of our $(1)$ and $(2)$ for fixed variables. Then this equation can be written as $$n=\frac{k(r^2-bm)+bm}{mr}$$ where $k$ is a function. Here are some values of $n$ for several values of $k$, expressed as $(k,n)$: $(1,r/m),(0,b/r),((r^2+bm)/(bm),r^3/(bm^2))$
From the first equation, $n = r/m$ (unless $m=0$ as a polynomial). This is a polynomial if and only if $m$ divides $r$. You don't need the second equation unless $m=0$.