To solve for the values of [tex]\( g \)[/tex] and [tex]\( h \)[/tex], we need to set up and solve a system of linear equations based on the information provided.
1. Kyle's total number of students can be represented by:
[tex]\[
8g + 6h = 62
\][/tex]
2. Lauren's total number of students can be represented by:
[tex]\[
5g + 10h = 70
\][/tex]
We can represent this system of linear equations in matrix form as follows:
[tex]\[
\begin{array}{l}
8g + 6h = 62 \\
5g + 10h = 70 \\
\end{array}
\][/tex]
In matrix form, it looks like this:
[tex]\[
\left[\begin{array}{cc}
8 & 6 \\
5 & 10
\end{array}\right]
\left[\begin{array}{l}
g \\
h
\end{array}\right]
= \left[\begin{array}{l}
62 \\
70
\end{array}\right]
\][/tex]
To solve this, the correct matrix that, when multiplied by the inverse of the coefficient matrix, will give us the values for [tex]\( g \)[/tex] and [tex]\( h \)[/tex]. Hence, the inverse of the matrix [tex]\(\left[\begin{array}{cc}
8 & 6 \\
5 & 10
\end{array}\right]\)[/tex] is needed. After calculating the inverse matrix and applying it, we get:
[tex]\[
\left[\begin{array}{l}
g \\
h
\end{array}\right]
= \left[\begin{array}{cc}
0.2 & -0.12 \\
-0.1 & 0.16
\end{array}\right]
\left[\begin{array}{l}
62 \\
70
\end{array}\right]
\][/tex]
Therefore, the correct answer is:
A. [tex]\(\left[\begin{array}{l}
g \\
h
\end{array}\right]=\left[\begin{array}{cc}
0.2 & -0.12 \\
-0.1 & 0.16
\end{array}\right]\left[\begin{array}{l}
62 \\
70
\end{array}\right]\)[/tex]
