To determine which matrix represents the given system of equations, let's break down the steps carefully.
We have the following system of equations:
[tex]\[
\begin{array}{c}
5x - y = 8 \\
2x - 3y = 12
\end{array}
\][/tex]
The goal is to form the augmented matrix corresponding to this system. An augmented matrix includes the coefficients of the variables and the constants from the right side of the equations, row by row.
### Step-by-Step Breakdown
1. Identify the coefficients and constants:
- For the first equation [tex]\( 5x - y = 8 \)[/tex]:
- The coefficient of [tex]\( x \)[/tex] is [tex]\( 5 \)[/tex].
- The coefficient of [tex]\( y \)[/tex] is [tex]\( -1 \)[/tex].
- The constant term is [tex]\( 8 \)[/tex].
- For the second equation [tex]\( 2x - 3y = 12 \)[/tex]:
- The coefficient of [tex]\( x \)[/tex] is [tex]\( 2 \)[/tex].
- The coefficient of [tex]\( y \)[/tex] is [tex]\( -3 \)[/tex].
- The constant term is [tex]\( 12 \)[/tex].
2. Form the augmented matrix:
- The first row will be [tex]\( [5, -1, 8] \)[/tex] (coefficients of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] from the first equation and the constant).
- The second row will be [tex]\( [2, -3, 12] \)[/tex] (coefficients of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] from the second equation and the constant).
Thus, the augmented matrix for the given system of equations is:
[tex]\[
\left[\begin{array}{ccc}
5 & -1 & 8 \\
2 & -3 & 12
\end{array}\right]
\][/tex]
Looking at the provided options:
A. [tex]\(\left[\begin{array}{ccc}
5 & -1 & 8 \\
2 & -3 & 12
\end{array}\right]\)[/tex]
B. [tex]\(\left[\begin{array}{lll}
2 & -1 & 8 \\
5 & -3 & 12
\end{array}\right]\)[/tex]
C. [tex]\(\left[\begin{array}{ccc}
5 & -1 & 12 \\
-2 & -3 & 8
\end{array}\right]\)[/tex]
D. [tex]\(\left[\begin{array}{lll}
5 & -3 & 8 \\
2 & -1 & 12
\end{array}\right]\)[/tex]
Only option A matches the matrix [tex]\(\left[\begin{array}{ccc}5 & -1 & 8 \\ 2 & -3 & 12\end{array}\right]\)[/tex].
Therefore, the correct matrix representing the given system of equations is:
[tex]\[
\boxed{\left[\begin{array}{ccc}5 & -1 & 8 \\ 2 & -3 & 12\end{array}\right]}
\][/tex]
