I have a system of $9$ linear equations in $9$ variables:

\begin{array}{rl}
-c\_{1}x\_{1} + x\_{2} + x\_{3} + x\_{4} + x\_{5} + x\_{6} + x\_{7} + x\_{8} + x\_{9} &= 0 \\
x\_{1} - c\_{2}x\_{2} + x\_{3} + x\_{4} + x\_{5} + x\_{6} + x\_{7} + x\_{8} + x\_{9} &= 0 \\
x\_{1} + x\_{2} - c\_{3}x\_{3} + x\_{4} + x\_{5} + x\_{6} + x\_{7} + x\_{8} + x\_{9} &= 0 \\
x\_{1} + x\_{2} + x\_{3} - c\_{4}x\_{4} + x\_{5} + x\_{6} + x\_{7} + x\_{8} + x\_{9} &= 0 \\
x\_{1} + x\_{2} + x\_{3} + x\_{4} - c\_{5}x\_{5} + x\_{6} + x\_{7} + x\_{8} + x\_{9} &= 0 \\
x\_{1} + x\_{2} + x\_{3} + x\_{4} + x\_{5} - c\_{6}x\_{6} + x\_{7} + x\_{8} + x\_{9} &= 0 \\
x\_{1} + x\_{2} + x\_{3} + x\_{4} + x\_{5} + x\_{6} - c\_{7}x\_{7} + x\_{8} + x\_{9} &= 0 \\ 
x\_{1} + x\_{2} + x\_{3} + x\_{4} + x\_{5} + x\_{6} + x\_{7} - c\_{8}x\_{8} + x\_{9} &= 0 \\ 
x\_{1} + x\_{2} + x\_{3} + x\_{4} + x\_{5} + x\_{6} + x\_{7} + x\_{8} - c\_{9}x\_{9} &= 0 
\end{array}

I want to find a general non-trivial solution for it. What would be the easiest and least time consuming way to find it by hand? I don't have a lot of background in maths, so I would very much appreciate if you actually found the solution and explained briefly.

Thanks in advance!

**EDIT:** Very important to mention is that always any $c\_{i} > 1$ and any $x\_{i} \geq 20$. Also it would be nice if someone posted how would a general non-trivial solution look in the form of $$S = \left \{( x\_{1}, x\_{2}, x\_{3}, x\_{4}, x\_{5}, x\_{6}, x\_{7}, x\_{8}, x\_{9}\right )\}$$