I am stuck on a question that i'm doing on gaussian elimination.
Any help is appreciated!
Hello, kaya2345!
Solve the following system of equations using Gaussian Elimination without pivots.
Not sure what that means . . .
. . $\displaystyle \begin{array}{ccc}6x_1+7x_2+8x_3 \:=\:21 \\ 7x_1+8x_2+9x_3 \:=\:24 \\ 8x_1+9x_2+9x_3 \:=\:26 \end{array}$
We have: .$\displaystyle \left[\begin{array}{ccc|c} 6&7&8&21 \\ 7&8&9&24 \\ 8&9&9&26 \end{array}\right]$
$\displaystyle \begin{array}{c}\\ R_2-R_1 \\ R_3-R_2 \end{array}\left[\begin{array}{ccc|c} 6&7&8&21 \\ 1&1&1&3 \\ 1&1&0&2 \end{array}\right]$
$\displaystyle \begin{array}{c}R_1-6R_2 \\ R_2-R_3 \\ \\ \end{array} \left[\begin{array}{ccc|c}0&1&2&3 \\ 0&0&1&1 \\ 1&1&0&2 \end{array}\right]$
$\displaystyle \begin{array}{c} \\ \\ R_3-R_1\end{array} \left[\begin{array}{ccc|c}0&1&2&3 \\ 0&0&1&1 \\ 1&0&\text{-}2&\text{-}1 \end{array}\right]$
$\displaystyle \begin{array}{c} R_1-2R_2 \\ \\ R_3+2R_2 \end{array} \left[\begin{array}{ccc|c}0&1&0&1 \\ 0&0&1&1 \\ 1&0&0&1 \end{array}\right]$
Therefore: .$\displaystyle \begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix} \;=\;\begin{bmatrix}1\\1\\1 \end{bmatrix}$
What's that nonsense about four decimal places?!
And since the roots are equal,
. . who cares where the pivots are?
This is an examination or coursework question so you have to be aware that you are being tested on the use of a particular method, not on whether you can simply solve the system. You might use some method that gets you the exact solution, but if it isn't the method being asked for you will earn no marks for your solution.
Gaussian elimination without any pivoting requires you to take multiples of the first equation from the second and third equations to so as to eliminate the first of the unknowns from those equations. Those multiples will be 7/6 and 8/6, and you work to the stated degree of accuracy. You then take a multiple of the new second equation from the third equation so as to remove the second of the unknowns. The third of the unknowns can then be calculated and back substitution gets you the other two. The results will naturally be approximations to the exact values.