Let $w,y, z \in R^n$ be vectors and $\sum_1^n w_i = 1$.
Suppose that $z$ and $w$ are both known and we are trying to solve for $y$.
It is given that $$z = \frac{y}{w^t \cdot y}$$ is it possible to solve for $y$?
As far as I could get:
$$(w^t \cdot y )z = y$$
$$w^t \cdot y \cdot z = y$$
Then let $D$ be a diagonal matrix with entries $w_1, ..., w_n$ and zeros everywhere else.
$$z^t \cdot D \cdot y = y$$
which implies that $z D = \mathbb{I}_n$, which is obviously wrong because the dimensions of $z D$ are (1 x n)(n x n) = (1 x n), which does not agree with the right hand side.
It is not possible to determine $y$. From $z=\frac y{w^t\cdot y}$ we can see that we can multiply $y$ by any scalar without changing $z$. In fact we can multiply both sides by $w^t$ and find $w^t\cdot z=1$. If that is true, $y$ can be any vector proportional to $z$. If it is not true, there is no solution.