To write the augmented matrix for the given system of linear equations, follow these steps:
1. Identify the coefficients of each variable in the system of equations.
2. Construct the coefficient matrix based on these coefficients.
3. Add the right-hand side constants of the equations as the last column in the matrix to form the augmented matrix.
Here are the equations provided:
[tex]\[
\begin{array}{l}
-9w + x + 8y + 2z = -6 \\
-5x + 9y - 7z = 6 \\
-3w - 6x + 4y + 9z = 4 \\
-3w - 8x + y = -2 \\
\end{array}
\][/tex]
To construct the coefficient matrix:
1. List the coefficients for each variable [tex]\( w, x, y, z \)[/tex] from each equation.
[tex]\[
\begin{pmatrix}
-9 & 1 & 8 & 2 \\
0 & -5 & 9 & -7 \\
-3 & -6 & 4 & 9 \\
-3 & -8 & 1 & 0 \\
\end{pmatrix}
\][/tex]
2. Include the right-hand side constants as the last column in the matrix.
[tex]\[
\begin{pmatrix}
-9 & 1 & 8 & 2 & -6 \\
0 & -5 & 9 & -7 & 6 \\
-3 & -6 & 4 & 9 & 4 \\
-3 & -8 & 1 & 0 & -2 \\
\end{pmatrix}
\][/tex]
Therefore, the augmented matrix for the given system of linear equations is:
[tex]\[
\begin{pmatrix}
-9 & 1 & 8 & 2 & -6 \\
0 & -5 & 9 & -7 & 6 \\
-3 & -6 & 4 & 9 & 4 \\
-3 & -8 & 1 & 0 & -2 \\
\end{pmatrix}
\][/tex]
