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Math Help - Gaussian Elimination Method

  1. #1
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    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
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    I'll do the easier one, the other is similar.
    Quote Originally Posted by flippin4u View Post
    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:

    .... x.... y..... z
    \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).

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

    Step 3: Read off your solutions:

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


    Now try the first one
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  3. #3
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    Hello, Kasey!

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


    2)\;\begin{array}{ccc}x - y + z & = & 17 \\<br />
x + y - z & = & \text{-}11 \\<br />
x - y - z & = & 9\end{array}

    We have: . \begin{bmatrix}1 & \text{-}1 & 1 &|& 17 \\<br />
1 & 1 & \text{-}1 &| &\text{-}11 \\<br />
1 & \text{-}1 & \text{-}1 &|& 9 \end{bmatrix}

    . \begin{array}{c}\\ R_2-R_1 \\ R_3-R_1\end{array}\;<br />
\begin{bmatrix}1 & \text{-}1 & 1 &|& 17 \\<br />
0 & 2 & \text{-}2 &|& \text{-}28 \\<br />
0 & 0 & \text{-}2 &|& \text{-}8 \end{bmatrix}

    . . . \begin{array}{c} \\ \frac{1}{2}R_2 \\ \text{-}\frac{1}{2}R_3\end{array}\;<br />
\begin{bmatrix}1 & \text{-}1 & 1 &|& 17 \\<br />
0 & 1 & \text{-}1 &|& \text{-}14 \\<br />
0 & 0 & 1 &|& 4 \end{bmatrix}

    . \begin{array}{c}R_1+R_2 \\ R_2+R_3 \\ \\ \end{array}\;<br />
\begin{bmatrix}1 & 0 & 0 &|& 3 \\<br />
0 & 1 & 0 &|& \text{-}10 \\<br />
0 & 0 & 1 &|& 4 \end{bmatrix}

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