# Thread: System of linear equations

1. ## 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.

2. What methods have you been taught or are you allowed to use?

3. 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.

4. 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.

5. 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.
$\displaystyle
\begin{bmatrix}1&2&-1&2&1\\1&2&2&0&1\\2&4&-2&3&1\end{bmatrix}\Rightarrow R_1-R_2 \ \mbox{and} \ -2R_1+R3\Rightarrow\begin{bmatrix}1&2&-1&2&1\\0&0&3&2&0\\0&0&0&-1&-1\end{bmatrix}$

$\displaystyle
\begin{bmatrix}1&2&-1&2&1\\0&0&3&2&0\\0&0&0&-1&-1\end{bmatrix}\Rightarrow\frac{1}{3}R_2, \ \ R_1-R3, \ \mbox{and} \ -1R_3\Rightarrow\begin{bmatrix}1&2&-1&1&0\\0&0&1&\frac{2}{3}&0\\0&0&0&1&1\end{bmatri x}$

Spoiler:

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

$\displaystyle y$

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

$\displaystyle s=-t$

$\displaystyle t$

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