Say we have a system of $m$ equations of the form: $$a_{11} x_1 + a_{12} x_2 + ... + a_{1n} x_n = p_1$$ $$...$$ $$a_{m1} x_1 + a_{m2} x_2 + ... + a_{mn} x_n = p_m$$ Where the $p_i,x_j \in \mathbb{R}$, but the $a_{ij}$'s are boolean values, i.e $a_{ij}\in\{0,1\}$.
So, if I want to solve for $\{a_{ij}\}$ and $\vec{x}$ for a given $\vec{p}$, a total of $(m+1)n$ variables,
- What are the conditions on $m$ ensuring the existence of at most one solution?
- Does it change anything if $p_i,x_j \in \mathbb{Z}$ instead of $\mathbb{R}$?
Here is a generic construction for this problem. Let $\vec{x} = \vec{1}$ and set all $a_{ij}=0$ except $a_{ii} = p_i$. That is a solution.
Note that this is an integer solution as well, as long as $\vec{p}$ is an integer vector.
I am surprised how you set up the problem though - typically, don't we have $a_{ij}$ fixed?
EDIT Even more than that. For any non-zero vector $\vec{x}$ there is a solution which works for any $\vec{p}$. Let $a_{ij} = 0$ when $i \neq j$ and let $a_{ii} = p_i/x_i$.