# Thread: Solving purely symbolic systems of linear equations

1. ## 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:

$u_i=\alpha_1+\alpha_2 x_i + \alpha_3 y_i$
$u_j=\alpha_1+\alpha_2 x_j + \alpha_3 y_j$
$u_m=\alpha_1+\alpha_2 x_m + \alpha_3 y_m$

I'm trying to solve for $\alpha_1, \alpha_2, \alpha_3$.

First I tried substitution by solving the first equation for $\alpha_1$ and plugging into the second equation. Then I tried solving that for $\alpha_2$, etc. The math just got so messy that I gave up.

I next tried setting up the equations in matrix format:

$

\left$\begin{matrix}1&x_i&y_i\\1&x_j&y_j\\1&x_m&y_m\end{ matrix}\right$

$

2. $a_{11}x_1+a_{12}x_2+ . . . + a_{1n}x_n=b_1$
$a_{21}x_1+a_{22}x_2+ . . . + a_{2n}x_n=b_2$
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
$a_{m1}x_1+a_{m2}x_2+ . . . + a_{mn}x_n=b_m$

do for that one u can find determinant and say if this then that, or if something it would be .... and so on...
or u can use substitution and show how does it work, and in which cases it will have one, more or none solutions ....

P.S. Sorry, but i didn't see correct... your $x, y$ are known members ???