Augmented Matrix

This is a type of Matrix where we have solutions to a System of Linear Equations.
$\begin{matrix} a_{11} x_{1} + a_{12} x_{2} + . . + a_{1 n} x_{n} = b_{1} \\ a_{21} x_{1} + a_{22} x_{2} + . . + a_{2 n} x_{n} = b_{2} \\ \dots \\ a_{m 1} x_{1} + a_{m 2} x_{2} + . . + a_{m n} x_{n} = b_{m} \end{matrix}$
Our augmented matrix looks like: $[\begin{array}{ccccc} a_{11} & a_{12} & \dots & a_{1 n} & b_{1} \\ a_{21} & a_{22} & \dots & a_{2 n} & b_{2} \\ \dots & \dots & \dots & \dots & \dots \\ a_{m 1} & a_{m 2} & \dots & a_{m n} & b_{m} \end{array}]$
Example:
- Create an augmented matrix from the following System of Linear Equations:
  - $\begin{matrix} x - 2 y + z = 1 \\ x + 6 y - 4 z = 4 \end{matrix}$
  - $[\begin{array}{cccc} 1 & - 2 & 1 & 1 \\ 1 & 6 & - 4 & 4 \end{array}]$
- Create a $2 \times 4$ system and augmented matrix:
  - $\begin{matrix} x + 4 y - 8 z + w = 10 \\ 7 x - 2 y + 2 z - w = 1 \end{matrix}$
  - $[\begin{array}{ccccc} 1 & 4 & - 8 & 1 & 10 \\ 7 & - 2 & 2 & - 1 & 1 \end{array}]$