1. ## Gaussian Elimination Method

Help me if you can: Here are the questions.

1) Sovle the system of equations by the Gaussian elimination method:

{ 2x + y - 3z = 1
{ 3x - y + 4z = 6
{ x + 2y - z = 9

2) Solver the system of equations by the Gaussian elimination method:

{ x - y + z = 17
{ x + y - z = -11
{ x - y - z = 9

Thanks in advance for any help!!!

Kasey

2. I'll do the easier one, the other is similar.
Originally Posted by flippin4u
2) Solver the system of equations by the Gaussian elimination method:

{ x - y + z = 17
{ x + y - z = -11
{ x - y - z = 9
Step 1: Create an augmented matrix:

....$\displaystyle x$....$\displaystyle y$.....$\displaystyle z$
$\displaystyle \left( \begin{array}{ccc|c} 1 & -1 & 1 & 17 \\ 1 & 1 & -1 & -11 \\ 1 & -1 & -1 & 9 \end{array} \right)$

Step 2:
Now, Run Gauss-Jordan elimination on it to bring it to row-echelon form (or reduced row echelon form, i will do this).

....$\displaystyle x$....$\displaystyle y$.....$\displaystyle z$
$\displaystyle \left( \begin{array}{ccc|c} 1 & -1 & 1 & 17 \\ 1 & 1 & -1 & -11 \\ 1 & -1 & -1 & 9 \end{array} \right)$
----------------------
$\displaystyle \left( \begin{array}{ccc|c} 1 & -1 & -1 & 9 \\ 0 & 0 & 2 & 8 \\ 0 & 2 & 0 & -20 \end{array} \right)$
----------------------
$\displaystyle \left( \begin{array}{ccc|c} 1 & -1 & -1 & 9 \\ 0 & 0 & 2 & 8 \\ 0 & 2 & 0 & -20 \end{array} \right)$
----------------------
$\displaystyle \left( \begin{array}{ccc|c} 1 & -1 & -1 & 9 \\ 0 & 1 & 0 & -10 \\ 0 & 0 & 1 & 4 \end{array} \right)$
----------------------
$\displaystyle \left( \begin{array}{ccc|c} 1 & 0 & -1 & -1 \\ 0 & 1 & 0 & -10 \\ 0 & 0 & 1 & 4 \end{array} \right)$
--------------------
$\displaystyle \left( \begin{array}{ccc|c} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & -10 \\ 0 & 0 & 1 & 4 \end{array} \right)$
------------------

$\displaystyle x = 3, y = -10 \mbox { and } z = 4$

Now try the first one

3. Hello, Kasey!

Do you anything about Gaussian elimination?
. . The second one practically falls apart . . .

$\displaystyle 2)\;\begin{array}{ccc}x - y + z & = & 17 \\ x + y - z & = & \text{-}11 \\ x - y - z & = & 9\end{array}$

We have: .$\displaystyle \begin{bmatrix}1 & \text{-}1 & 1 &|& 17 \\ 1 & 1 & \text{-}1 &| &\text{-}11 \\ 1 & \text{-}1 & \text{-}1 &|& 9 \end{bmatrix}$

. $\displaystyle \begin{array}{c}\\ R_2-R_1 \\ R_3-R_1\end{array}\; \begin{bmatrix}1 & \text{-}1 & 1 &|& 17 \\ 0 & 2 & \text{-}2 &|& \text{-}28 \\ 0 & 0 & \text{-}2 &|& \text{-}8 \end{bmatrix}$

. . . $\displaystyle \begin{array}{c} \\ \frac{1}{2}R_2 \\ \text{-}\frac{1}{2}R_3\end{array}\; \begin{bmatrix}1 & \text{-}1 & 1 &|& 17 \\ 0 & 1 & \text{-}1 &|& \text{-}14 \\ 0 & 0 & 1 &|& 4 \end{bmatrix}$

. $\displaystyle \begin{array}{c}R_1+R_2 \\ R_2+R_3 \\ \\ \end{array}\; \begin{bmatrix}1 & 0 & 0 &|& 3 \\ 0 & 1 & 0 &|& \text{-}10 \\ 0 & 0 & 1 &|& 4 \end{bmatrix}$