Solving purely symbolic systems of linear equations

Hello,

I'm working through a finite element text which provides a symbolic solution to a system of linear equations which I suspect might be incorrect based on some checks that I've done. I'm interested in double checking their math, but I'm finding it excessively complicated with the techniques that I know how to use. Here is the system:

I'm trying to solve for .

First I tried substitution by solving the first equation for and plugging into the second equation. Then I tried solving that for , etc. The math just got so messy that I gave up.

I next tried setting up the equations in matrix format: