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:

I tried solving for the alpha vector by inverting the 3x3 matrix using Gauss-Jordan elimination on the following:

This seems just as complicated. I noticed that the book made mention of a determinant in the solution (although it did not show a solution process. Is there an easier way of solving this system symbolically that I am not aware of?

Thanks,

James