To determine the augmented matrix for the given system of linear equations, we must first understand that an augmented matrix is a compact representation where the coefficients of the variables and the constants on the right-hand side of the equations are included in one matrix.
Given the system of equations:
[tex]\[
\begin{array}{l}
-4x + 3y = 216 \\
10x - 4y = -156
\end{array}
\][/tex]
We write down the coefficients of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] from both equations along with the constants on the right side in a matrix form.
For the first equation, [tex]\( -4x + 3y = 216 \)[/tex]:
- The coefficient of [tex]\( x \)[/tex] is [tex]\( -4 \)[/tex].
- The coefficient of [tex]\( y \)[/tex] is [tex]\( 3 \)[/tex].
- The constant term is [tex]\( 216 \)[/tex].
Therefore, the first row of the augmented matrix will be:
[tex]\[ [-4, 3, 216] \][/tex]
For the second equation, [tex]\( 10x - 4y = -156 \)[/tex]:
- The coefficient of [tex]\( x \)[/tex] is [tex]\( 10 \)[/tex].
- The coefficient of [tex]\( y \)[/tex] is [tex]\( -4 \)[/tex].
- The constant term is [tex]\( -156 \)[/tex].
Therefore, the second row of the augmented matrix will be:
[tex]\[ [10, -4, -156] \][/tex]
Combining these rows, we form the complete augmented matrix:
[tex]\[
\left[\begin{array}{ccc}
-4 & 3 & 216 \\
10 & -4 & -156
\end{array}\right]
\][/tex]
Thus, the correct augmented matrix representing the given system of equations is:
[tex]\[
\left[\begin{array}{ccc}-4 & 3 & 216 \\ 10 & -4 & -156\end{array}\right]
\][/tex]
