How do you solve using gaussian elimination or gauss-jordan elimination, X- 3Y + 2Z = -5, 4X - 11Y + 4Z = -7, 3X - 8Y + 2Z = -2?

Sep 26, 2017

The solution is $\left(\begin{matrix}X \\ Y \\ Z\end{matrix}\right) = \left(\begin{matrix}34 + 10 Z \\ 13 + 4 Z \\ Z\end{matrix}\right)$

Explanation:

Perform the gauss-jordan elimination on the augmented matrix

$\left(\begin{matrix}1 & - 3 & 2 & | & - 5 \\ 4 & - 11 & 4 & | & - 7 \\ 3 & - 8 & 2 & | & - 2\end{matrix}\right)$

We perform the following operations on the rows

$R 2 \leftarrow R 2 - 4 R 1$ and $R 3 \leftarrow R 3 - 3 R 1$

$\left(\begin{matrix}1 & - 3 & 2 & | & - 5 \\ 0 & 1 & - 4 & | & 13 \\ 0 & 1 & - 4 & | & 13\end{matrix}\right)$

$R 3 \leftarrow R 3 - R 2$ and $R 1 \leftarrow R 1 + 3 R 2$

$\left(\begin{matrix}1 & 0 & - 10 & | & 34 \\ 0 & 1 & - 4 & | & 13 \\ 0 & 0 & 0 & | & 0\end{matrix}\right)$

The solution is

$\left(\begin{matrix}X \\ Y \\ Z\end{matrix}\right) = \left(\begin{matrix}34 + 10 Z \\ 13 + 4 Z \\ Z\end{matrix}\right)$