# Solving purely symbolic systems of linear equations

• Jul 31st 2010, 11:50 PM
jfortiv
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:

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

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

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

I next tried setting up the equations in matrix format:

$\displaystyle \left$\begin{matrix}1&x_i&y_i\\1&x_j&y_j\\1&x_m&y_m\end{ matrix}\right$$
• Aug 1st 2010, 12:00 AM
yeKciM
$\displaystyle a_{11}x_1+a_{12}x_2+ . . . + a_{1n}x_n=b_1$
$\displaystyle a_{21}x_1+a_{22}x_2+ . . . + a_{2n}x_n=b_2$
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
$\displaystyle a_{m1}x_1+a_{m2}x_2+ . . . + a_{mn}x_n=b_m$

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

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