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Math Help - System of linear equations

  1. #1
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    Question System of linear equations

    Find the basic solutions and write the general solution as a linear combination of the basic solutions.
    x+2y-z+2s+t=0
    x+2y+2z+t=0
    2x+4y-2z+3s+t=0
    Thanks alot.
    Last edited by mr fantastic; January 6th 2011 at 05:38 PM. Reason: Deleted begging in title.
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  2. #2
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    What methods have you been taught or are you allowed to use?
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  3. #3
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    Looks like you're going to want to use some form of Gauss-Jordan Reduction. Why don't you try to start it off so we can see where you get stuck.
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  4. #4
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    You don't really need to use anything as sophisticated as "Gauss-Jordan". For example, subtracting the first two equations immediately eliminates x and y and subtracting twice the second equation from the third does the same thing, leaving two equations to be solved for z, s, and t.
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  5. #5
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    Quote Originally Posted by seit View Post
    Find the basic solutions and write the general solution as a linear combination of the basic solutions.
    x+2y-z+2s+t=0
    x+2y+2z+t=0
    2x+4y-2z+3s+t=0
    Thanks alot.
    \displaystyle<br />
\begin{bmatrix}1&2&-1&2&1\\1&2&2&0&1\\2&4&-2&3&1\end{bmatrix}\Rightarrow R_1-R_2 \ \mbox{and} \ -2R_1+R3\Rightarrow\begin{bmatrix}1&2&-1&2&1\\0&0&3&2&0\\0&0&0&-1&-1\end{bmatrix}

    \displaystyle<br />
\begin{bmatrix}1&2&-1&2&1\\0&0&3&2&0\\0&0&0&-1&-1\end{bmatrix}\Rightarrow\frac{1}{3}R_2, \ \ R_1-R3, \ \mbox{and} \ -1R_3\Rightarrow\begin{bmatrix}1&2&-1&1&0\\0&0&1&\frac{2}{3}&0\\0&0&0&1&1\end{bmatri  x}
    Spoiler:

    \displaystyle x=-2y+z-s-t

    \displaystyle y

    \displaystyle z=-s\frac{2}{3}

    \displaystyle s=-t

    \displaystyle t

    \displaysytle<br />
\begin{bmatrix}-2y +z-s-t\\ y\\ -s\frac{2}{3}\\ -t\\ t\end{bmatrix}\Rightarrow y\begin{bmatrix}-2\\1\\0\\0\\0\end{bmatrix}+z\begin{bmatrix}1\\0\\0  \\0\\0\end{bmatrix}+s\begin{bmatrix}-1\\0\\ -\frac{2}{3}\\0\\0\end{bmatrix}+t\begin{bmatrix}-1\\0\\0\\-1\\1\end{bmatrix}
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