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.
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.
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.
$\displaystyle \displaystyle
\begin{bmatrix}1&2&-1&2&1\\1&2&2&0&1\\2&4&-2&3&1\end{bmatrix}\Rightarrow R_1-R_2 \ \mbox{and} \ -2R_1+R3\Rightarrow\begin{bmatrix}1&2&-1&2&1\\0&0&3&2&0\\0&0&0&-1&-1\end{bmatrix}$
$\displaystyle \displaystyle
\begin{bmatrix}1&2&-1&2&1\\0&0&3&2&0\\0&0&0&-1&-1\end{bmatrix}\Rightarrow\frac{1}{3}R_2, \ \ R_1-R3, \ \mbox{and} \ -1R_3\Rightarrow\begin{bmatrix}1&2&-1&1&0\\0&0&1&\frac{2}{3}&0\\0&0&0&1&1\end{bmatri x}$
Spoiler: