To write the augmented matrix for the given system of equations, we first need to ensure that each equation is in the standard form [tex]\(Ax + By + Cz = D\)[/tex].
First, let's align all the equations properly:
1. [tex]\(7x + 9 = -7z\)[/tex]
Rearrange to:
[tex]\[
7x + 7z = -9
\][/tex]
2. [tex]\(-4x + 4z = -8 - 6y\)[/tex]
Rearrange to:
[tex]\[
-4x - 6y + 4z = -8
\][/tex]
3. [tex]\(-3x - 7y + 9z = 2\)[/tex]
Now, let's put these equations into matrix form. We do this by taking the coefficients of [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] and placing them into a matrix, and then creating an augmented matrix with the constants on the right side of the equations.
[tex]\[
\begin{array}{ccc|c}
7 & 0 & 7 & -9 \\
-4 & -6 & 4 & -8 \\
-3 & -7 & 9 & 2 \\
\end{array}
\][/tex]
With this information, the correct matrix representation of the system is:
[tex]\[
\left[\begin{array}{rrr|r}
7 & 0 & 7 & -9 \\
-4 & -6 & 4 & -8 \\
-3 & -7 & 9 & 2
\end{array}\right]
\][/tex]
So, the correct answer is:
a.
[tex]\(\left[\begin{array}{rrr|r}7 & 0 & 7 & -9 \\ -4 & 6 & 4 & -8 \\ -3 & -7 & 9 & 2\end{array}\right]\)[/tex]
