Answer
$[A|B]=\left[\begin{array}{rrr|r}
{1}&{1}&{-1}&{2}\\
{3}&{-2}&{0}&{2}\\
{5}&{3}&{-1}&{1}\end{array}\right]$
Work Step by Step
The standard form of a linear equation can be expressed as:
$$a_{i1}x_{1}+a_{i2}x_{2}+...+a_{in}x_{n}=b_{i}$$
where the index $i$ indicates that it is the i-th equation of a system of equations.
In order to write the augmented matrix $[A|B]$ of a system of equations in standard form, we must follow some important points:
1. To express a system in matrix form, we must extract the coefficients of the variables and constants.
2. Draw a vertical line to separate the coefficient entries from the constants (essentially replacing the equal signs).
3. The entries of the coefficient matrix $A=[a_{ij}]$ must be placed to the left of the lline.
4. The constants of the $B=[b_{i}]$ must be placed to the right of the line.
We rewrite the system in standard form as: $x+y-z=2 \\3x-2y=2 \\5x+3y-z=1$
Now, we have: $ A=\left[\begin{array}{rrrr}
1 & 1 & -1\\
3 & -2 & 0\\
5 & 3 & -1
\end{array}\right]$ and $B=\left[\begin{array}{l}
2\\
2\\
1
\end{array}\right]$
Finally, we can write the system as an augmented matrix $[A|B]$ as follows: $[A|B]=\left[\begin{array}{rrr|r}
{1}&{1}&{-1}&{2}\\
{3}&{-2}&{0}&{2}\\
{5}&{3}&{-1}&{1}\end{array}\right]$
