Solving system of equations in rationals

Question

Do there exist solutions to solve system of $n-2$ equations with $n-2$ variables where $n$ is fixed even integer and $a_i,b,c\in\mathbb{Q},i\in\{0,1,2,\cdots,n-5\}$ $$\left\{ \begin{array}{ll} a_{n-5}+b=\frac{{n\choose 2}}{n}\\ a_{n-6}+ba_{n-5}+c=\frac{{n\choose 3}}{n}\\ a_{n-7}+ba_{n-6}+ca_{n-5}=\frac{{n\choose 4}}{n}\\a_{n-8}+ba_{n-7}+ca_{n-6}=\frac{{n\choose 5}}{n}\\\vdots\\a_0+ba_1+ca_2=\frac{{n\choose n-3}}{n}\\ba_0+ca_1=\frac{{n\choose n-2}}{n}\\ca_0=\frac{{n\choose n-1}}{n} \end{array} \right.$$ I stumbled upon this while solving some problem,I haven't worked with matrices in school and I couldn't find any material for solving system of equations,couldn't even find a good explanation for $2\times 2$ system by matrices.I can understand solution(google + wikipedia makes wonders),or if someone is to guide me how to begin I would highly appreciate it.