Solving a system of equations with as many unknown and known variables Consider the following four variables for each $j=1,\ldots,J$ and $J>1$:
$\theta_{j}\in\mathbb{R}$,
$\tilde{\theta}_{j}=\sum_{i=1}^{j}\theta_{i}$,
$a_{j}\in\mathbb{R}$,
$\tilde{a}_{j}=\sum_{i=1}^{j}a_{i}$,
Assume that all $\theta_{j},\tilde{\theta}_{j}$ are unobserved, while all $a_{j},\tilde{a}_{j}$ are observed.
Further, assume a model in which we restrict
\begin{align}
\frac{\theta_{j}}{a_{j}} &=\frac{\theta_{k}}{a_{k}},\quad\forall j,k, \\
\frac{\theta_{j}}{a_{j}} &=\frac{\tilde{\theta}_{k}}{\tilde{a}_{k}},\quad\forall j\leq k
\end{align}
(1) Would it be possible to solve the model for $\theta_{j}$ in terms of only $a_j$'s?
(2) If this is not possible, what kind of other restrictions would I need to place on the model? I have some freedom on placing restrictions on the $a_j$'s, but not on the $\theta_j$'s.
Thanks!
 A: With the clarifying comments on the Question, we can clear away some of the obscuring notation and direct attention to the central restrictions and the total set of solutions.  From this we find that the solutions have a single degree of freedom, so that one additional condition on the $\theta_i$ collectively suffices to determine them, given nonzero values of $\alpha_i$.
The assumption that $\alpha_i \neq 0$ is needed to make sense of the indicated divisions:
$$ \frac{\theta_j}{\alpha_j} = \frac{\theta_k}{\alpha_k}  \;\; \forall j,k $$
Similarly for the indicated division of the tilde (summation) variables, we ought to require the running sums $\tilde\alpha_k \neq 0$.  Such a condition would be automatically satisfied if it were known the $\alpha_j \gt 0$, but that is not stated in the Question (so presumably negative $\alpha_j$ are possible).
One would say that the equations above tell us there is a (real) constant of proportionality between the $\theta_j$ and $\alpha_j$ terms.  Taking that unknown constant to be $r\in \mathbb R$, all model solutions have the form:
$$ \theta_j = r \alpha_j $$
It follows that $\tilde\theta_k = \sum_{j=1}^k \theta_j$ would bear the same relationship to the $\tilde\alpha_k = \sum_{j=1}^k \alpha_j$, namely:
$$ \frac{\tilde\theta_k}{\tilde\alpha_k} = r \;\; \forall k $$
Thus the additional ratio conditions are satisfied.
In summary there is only a parameter $r$ that needs to be determined in order to find $\theta_j$ given $\alpha_j$, and that parameter can be any real number.  If not know directly, the value $r$ could be derived from any single value of $\theta_j$ or from a value of one of the running sums $\tilde\theta_k$.
There are additional ways one might impose an extra condition, and it should be left to the problem's poser to suggest what is most naturally imposed.
