To determine the augmented matrix for the given system of linear equations, we need to follow these steps:
1. Write the system of equations in matrix form:
The given system is:
[tex]\[
\begin{array}{l}
-4x + 3y = 216 \\
10x - 4y = -156
\end{array}
\][/tex]
2. Create the coefficient matrix:
This matrix includes the coefficients of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] from both equations:
[tex]\[
\left[\begin{array}{cc}
-4 & 3 \\
10 & -4
\end{array}\right]
\][/tex]
3. Include the constants from the right-hand side as an additional column:
Place these constants (216 and -156) as an additional column in the coefficient matrix. This forms the augmented matrix:
[tex]\[
\left[\begin{array}{ccc}
-4 & 3 & 216 \\
10 & -4 & -156
\end{array}\right]
\][/tex]
4. Identify the correct option:
The augmented matrix that corresponds to the given system of equations is:
[tex]\[
\left[\begin{array}{ccc}
-4 & 3 & 216 \\
10 & -4 & -156
\end{array}\right]
\][/tex]
So, the correct answer is:
[tex]\[
\left[\begin{array}{ccc}-4 & 3 & 216 \\ 10 & -4 & -156\end{array}\right]
\][/tex]
