To determine the coefficient matrix that represents the given system of linear equations, we need to identify the coefficients of the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in each equation.
The system of equations provided is:
[tex]\[
\begin{aligned}
7x + 8y &= 28 \\
-3x + 9y &= -24
\end{aligned}
\][/tex]
For a coefficient matrix, we only include the coefficients of the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex], and we do not include the constants (the numbers on the right side of the equations).
From the first equation [tex]\(7x + 8y = 28\)[/tex], the coefficients of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are [tex]\(7\)[/tex] and [tex]\(8\)[/tex], respectively.
From the second equation [tex]\(-3x + 9y = -24\)[/tex], the coefficients of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are [tex]\(-3\)[/tex] and [tex]\(9\)[/tex], respectively.
Thus, the coefficient matrix is constructed by placing these coefficients in their respective positions corresponding to each equation. The first row will consist of the coefficients of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] from the first equation, and the second row will consist of the coefficients of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] from the second equation.
Therefore, the coefficient matrix is:
[tex]\[ \left[\begin{array}{cc}
7 & 8 \\
-3 & 9
\end{array}\right] \][/tex]
From the given options, the matrix that accurately represents the coefficient matrix is:
[tex]\[ \left[\begin{array}{cc}
7 & 8 \\
-3 & 9
\end{array}\right] \][/tex]
So, the correct answer is:
[tex]\[ \left[\begin{array}{cc}
7 & 8 \\
-3 & 9
\end{array}\right] \][/tex]
