What free software can I use to solve a system of linear equations containing an unknown?

Try Sage. You can use it through a web browser even.

You can try octave. It is pretty handy and easy to use. It is an opensource version of MATLAB. It is not completely clear from your question, if you want to solve the linear system with $n$ being an unknown quantity/variable(i.e., treating $n$ as a symbol) or if you want to solve the linear system by letting $n$ take on different values. Either way, if you want to use $n$ as an unknown variable, then you should use syms to treat $n$ as a symbol.

You should give Sequalator a try. It is capable of solving thousands of Linear Simultaneous Equations with great precision and speed. On my mediocre laptop it is able to solve 1000 equations just below a second.

EDIT: I am the developer of Sequalator.