How can I solve systems of equations of the following form?

Question

I have the following system of equations that I would like to solve / find solutions for certain variables: $$ \begin{bmatrix} 1&&&&&&\newline -1&1&x_{1}&x_{2}&&&\newline &x_{3}&1&x_{4}&&&\newline &x_{5}&x_{6}&1&&&\newline &x_{7}&x_{8}&x_{9}&1&&\newline &x_{10}&x_{11}&x_{12}&&1&\newline &x_{13}&x_{14}&x_{15}&&&1\newline \end{bmatrix} % \begin{bmatrix} a\newline y_{1}\newline y_{2}\newline y_{3}\newline \frac{a}{3}\newline \frac{a}{3}\newline \frac{a}{3}\newline \end{bmatrix} = \begin{bmatrix} a\newline 0\newline 0\newline 0\newline 0\newline 0\newline 0\newline \end{bmatrix} $$ Specifically, I want to find solutions for {$x_{1},...,x_{15}$}, where $x_{i}∈ ${$-\frac{1}{2},0$}, $y_{i}>0$, and $a$ is some unknown constant. I'm not sure if this is possible to solve faster than brute force checking. Additionally, this should be generalizable to matrices where the number of $x$s per row and size of the matrix can be different. If this seems like an "ask to solve x, actually needs to solve y" problem, please let me know I will provide more details.