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Math Help - [SOLVED] Gauss-Jordan method

  1. #1
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    [SOLVED] Gauss-Jordan method

    Can someone please explain to me the Guass-Jordan method to solve a system of equations?

    My homework question is:

    Use the Gauss-Jordan method to solve the system of equations.
    4x-6y=54
    20x-30y=270

    I am sure that if someone would simplify the method for me (I guess explain it to me a little more clearly) I would be able to figure it out. I don't want the answer, I would like to figure it out on my own, but just need someone to lead me in the right direction.
    Last edited by wilsoac; February 18th 2009 at 03:53 PM.
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  2. #2
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    Hello.
    Using the Gauss-Jordan method you'll have to transform the augmented matrix using row elimination to transform the matrix into this form.
    <br />
\begin{pmatrix} a_{11} & 0 & \dots & 0 & \vdots & a_{1, n+1} \\ <br />
0 & a_{22} & \ddots & \vdots & \vdots & a_{2, n+1}\\<br />
\vdots & \ddots & \ddots & 0 & \vdots & \vdots \\<br />
0 & \dots & 0 & a_{nn} & \vdots & a_{n, n+1} \end{pmatrix}

    and the solutions will be
     x_i = \frac{a_{i, n+1}}{a_{ii}}

    here's one example

    2x_1-x_2=4
    x_1-x_2=1

    the augmented matrix is

    \begin{pmatrix} 2 & -1 & \vdots & 4 \\ 1 & -1 & \vdots & 1 \end{pmatrix}

    after performing these transformations:Row2-(1/2)*Row1 then after this, Row1-2*Row2
    the augmented matrix will be
    <br />
\begin{pmatrix} 2 & 0 & \vdots & 6 \\ 0 & -\frac{1}{2} & \vdots & -1 \end{pmatrix}

    the solutions will be: x_1=\frac{6}{2} = 3, x_2=\frac{-1}{\frac{-1}{2}}=2
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