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Give the parameterized solution for the dependent system represented by the matrix: $\left(\begin{array}{ccc|r} 1 & 0 & 3 & 6\\ 0 & 1 & 2 & -4\\ 0 & 0 & 0 & 0 \end{array}\right).$

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    also, is this homework?2011-02-07

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I LaTeX'ed your original post so that the matrix you are trying to convey is obvious. Seems like the matrix you are talking about is the following (got it from the 'edit' text):

$\begin{bmatrix} 1 & 0 & 3 & | & 6\\ 0 & 1 & 2 & | & -4\\ 0 & 0 & 0 & | & 0 \end{bmatrix}$

The way to go here is from the bottom to up. You see that the third coordinate can be anything (it's a free variable), since $x_{3}$ times $0$ equals $0$ in all cases (I am looking at the third coefficient of the third equation in your system). So let $x_{3}=t$. Then the second equation is of form $0x_{1}+1x_{2}+2t=-4$. So, $x_{2}=-4-2t$. And, finally, the first equation can be represented as $1x_{1}+0x_{2}+3t=6$, which amounts to $x_{1}=6-3t$. So your solution vector is $\begin{bmatrix} 6-3t\\ -4-2t\\ t \end{bmatrix}$.

P.S. You can use http://www.codecogs.com/latex/eqneditor.php to convert your math to LaTeX.