Answer
\[
\left[
\begin{array}{ccc|c}
4 & -2 & 3 &4 \\
3 & 5 & 1 &7\\5&-1&4&7
\end{array}
\right]
\]
Work Step by Step
We will write the given equations in such a form that the variables terms are on the left-hand side and the constants on the right-hand side.
Thus, we have:
$4x-2y+3z=4 \quad\quad \text{(Add 4 to both sides of the first equation.)}\\ 3x+5y+z=7 \quad\quad \text{(Add 7 from both sides of the first equation.)}\\5x-y+4z=7 \quad\quad \text{(Add 7 to both sides of the third equation.)}$
We transform the above equations into a $3 \times 4$ matrix such that the coefficients of the variables are on the left-hand side and the constants on the right-hand side of the vertical bar.
Therefore, the $3 \times 4$ matrix can be expressed as:
\[
\left[
\begin{array}{ccc|c}
4 & -2 & 3 &4 \\
3 & 5 & 1 &7\\5&-1&4&7
\end{array}
\right]
\]