System of linear equations

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January 6th 2011, 05:24 PM
seit

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.
January 6th 2011, 05:26 PM
pickslides

What methods have you been taught or are you allowed to use?
January 6th 2011, 05:53 PM
DrSteve

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.
January 7th 2011, 05:21 PM
HallsofIvy

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.
January 7th 2011, 06:09 PM
dwsmith

Quote:

Originally Posted by seit

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:

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

$\displaystyle y$

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

$\displaystyle s=-t$

$\displaystyle t$

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