Answer
$[A|B]=\left[\begin{array}{rrr|r}
{5 }&{-1} &{-1} &{0 }\\
{1 }&{1}&{0} &{5 }\\{2}&{0}&{-3}&{2}\end{array}\right]$
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: $5x-y-z=0 \\x+y=5 \\2x-3z=2$
Now, we have:
$ A=\left[\begin{array}{ll}
5 & -1 &-1\\
1 & 1&0 \\2&0&-3
\end{array}\right]$ and $B=\left[\begin{array}{l}
0\\
5\\2
\end{array}\right]$
Finally, we can write the system as an augmented matrix $[A|B]$ as follows:
$[A|B]=\left[\begin{array}{rrr|r}
{5 }&{-1} &{-1} &{0 }\\
{1 }&{1}&{0} &{5 }\\{2}&{0}&{-3}&{2}\end{array}\right]$
