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Dakarai wrote the system of linear equations below.

[tex]\[
\begin{aligned}
7x + 8y &= 28 \\
-3x + 9y &= -24
\end{aligned}
\][/tex]

Dakarai then wrote the coefficient matrix that represents this system. Which matrix could she have written?

A. [tex]\(\left[\begin{array}{cc}7 & 8 \\ -3 & 9\end{array}\right]\)[/tex]

B. [tex]\(\left[\begin{array}{cc}7 & -3 \\ 8 & 9\end{array}\right]\)[/tex]

C. [tex]\(\left[\begin{array}{lll}7 & 8 & 28\end{array}\right]\)[/tex]


Sagot :

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]