# How do you solve the system x+y-2z=5, x+2y+z=8, 2x+3y-z=13?

Mar 20, 2018

$x = 5 k + 2$, $y = 3 - 3 k$ and $z = k$

#### Explanation:

Perform the Gauss Jordan elimination on the augmented matrix

$A = \left(\begin{matrix}1 & 1 & - 2 & | & 5 \\ 1 & 2 & 1 & | & 8 \\ 2 & 3 & - 1 & | & 13\end{matrix}\right)$

I have written the equations not in the sequence as in the question in order to get $1$ as pivot.

Perform the folowing operations on the rows of the matrix

$R 2 \leftarrow R 2 - R 1$; $R 3 \leftarrow R 3 - 2 R 1$

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

Consequently this system has infinite solutions. After choosing $z = k$, $y$ must be $3 - 3 k$ and $x$ must be $5 k + 2$