To determine the augmented matrix for the given system of equations, let's first rewrite each equation in standard form, such that each equation is set equal to a constant:
1. [tex]\(3w - x = 2y + z - 4\)[/tex]
Rearranging gives:
[tex]\[3w - x - 2y - z = -4\][/tex]
2. [tex]\(9x - y + z = 10\)[/tex]
This equation is already in the correct form with no [tex]\(w\)[/tex] term:
[tex]\[0w + 9x - y + z = 10\][/tex]
3. [tex]\(4w + 3y - z = 7\)[/tex]
This equation is also in the correct form:
[tex]\[4w + 0x + 3y - z = 7\][/tex]
4. [tex]\(12x + 17 = 2y - z + 6\)[/tex]
Rearranging gives:
[tex]\[12x + 17 = 2y - z + 6\][/tex]
[tex]\[12x - 2y + z = -11\][/tex]
Here, there is no [tex]\(w\)[/tex] term.
Now, we can write the augmented matrix formed by the coefficients of [tex]\(w\)[/tex], [tex]\(x\)[/tex], [tex]\(y\)[/tex], [tex]\(z\)[/tex] and the constants on the right-hand side:
[tex]\[
\begin{pmatrix}
3 & -1 & -2 & -1 & -4 \\
0 & 9 & -1 & 1 & 10 \\
4 & 0 & 3 & -1 & 7 \\
0 & 12 & -2 & 1 & -11
\end{pmatrix}
\][/tex]
Examining the answer choices, the correct matrix corresponds to:
[tex]\[
\left[\begin{array}{rrrr|r}
3 & -1 & -2 & -1 & -4 \\
0 & 9 & -1 & 1 & 10 \\
4 & 0 & 3 & -1 & 7 \\
0 & 12 & -2 & 1 & -11
\end{array}\right]
\][/tex]
Therefore, the correct choice is:
B. [tex]\(\left.\left\lvert\, \begin{array}{rrrr|r}3 & -1 & -2 & -1 & -4 \\ 0 & 9 & -1 & 1 & 10 \\ 4 & 0 & 3 & -1 & 7 \\ 0 & 12 & -2 & 1 & -11\end{array}\right.\right]\)[/tex]
