Let $x_i,\ldots,x_n$ and $y_{i,j}$ where $1 \le i \le n$ and $1 \le j \le L$ be a set of variables in the following equations (and inequalities):
$\sum_{i=1}^n x_i y_{i,l} = c_l$ for $1 \le l \le L$
$\sum_{i=1}^n x_i = 1$
$\sum_{i=1}^n y_{i,l} = 1$ for $1 \le l \le L$
$x_i \ge 0, y_{i,l} \ge 0$ for all $1 \le i \le n$ and $1 \le l \le L$
where $c_l$ are non-negative constants.
Is there a solution to these equations if $L$ is large enough? Is the solution unique, and is there an analytic/algorithm way to find it?
Thanks!
(I had a problem finding the right category for this question. I wanted to put it as "algebra" or "equations" but neither exist? since the application is for something related to probability, I decided to use "probability".)