Answer
$[A|B]=\left[\begin{array}{rr|r}
{ 4/3 }&{-3/2 } &{ 3/4 }\\
{ -1/4 }&{ 1/3 } &{ 2/3 }\end{array}\right]$
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: $\dfrac{4}{3}x-\dfrac{3}{2}y=\dfrac{3}{4} \\\dfrac{-1}{4}x+\dfrac{1}{3}y=\dfrac{2}{3}$
Now, we have:
$ A=\left[\begin{array}{ll}
4/3 & -3/2\\
-1/4 & 1/3
\end{array}\right]$ and $B=\left[\begin{array}{l}
3/4\\
2/3
\end{array}\right]$
Finally, we can write the system as an augmented matrix $[A|B]$ as follows:
$[A|B]=\left[\begin{array}{rr|r}
{ 4/3 }&{-3/2 } &{ 3/4 }\\
{ -1/4 }&{ 1/3 } &{ 2/3 }\end{array}\right]$
