In the given problem, we need to represent the situation with a system of linear equations and then find the correct matrix equation that shows the solution to that system.
Let's denote:
- [tex]\( g \)[/tex] as the number of students in each of Kyle's classes.
- [tex]\( h \)[/tex] as the number of students in each of Lauren's classes.
According to the problem:
- Kyle has 8 classes with [tex]\( g \)[/tex] students in each class and 6 classes with [tex]\( h \)[/tex] students in each class, totaling 62 students.
- Lauren has 5 classes with [tex]\( g \)[/tex] students in each class and 10 classes with [tex]\( h \)[/tex] students in each class, totaling 70 students.
We can write this situation as a system of linear equations:
[tex]\[ 8g + 6h = 62 \][/tex]
[tex]\[ 5g + 10h = 70 \][/tex]
To find the solution to this system of equations using matrix representation, let’s define the matrix [tex]\( A \)[/tex] and the vector [tex]\( B \)[/tex]:
[tex]\[ A = \begin{bmatrix} 8 & 6 \\ 5 & 10 \end{bmatrix} \][/tex]
[tex]\[ B = \begin{bmatrix} 62 \\ 70 \end{bmatrix} \][/tex]
We need to find the values of [tex]\( g \)[/tex] and [tex]\( h \)[/tex] that satisfy [tex]\( A \mathbf{x} = B \)[/tex], where [tex]\( \mathbf{x} = \begin{bmatrix} g \\ h \end{bmatrix} \)[/tex].
The correct way to solve for [tex]\( \mathbf{x} \)[/tex] in the matrix equation [tex]\( A \mathbf{x} = B \)[/tex] is to multiply both sides by the inverse of matrix [tex]\( A \)[/tex]:
[tex]\[ \mathbf{x} = A^{-1} B \][/tex]
We now need to confirm which of the given options represents this solution setup correctly.
Option A:
[tex]\[ \left[\begin{array}{l}g \\ h\end{array}\right] = \left[\begin{array}{cc}0.16 & 0.12 \\ 0.1 & 0.2\end{array}\right]\left[\begin{array}{l}62 \\ 70\end{array}\right] \][/tex]
This would imply that the coefficients (0.16, 0.12, 0.1, 0.2) are obtained such that they correspond to the inverses of the relationships described by matrix [tex]\( A \)[/tex]. To check, let’s verify if these are the correct inverses, or one can compute directly.
Option B:
[tex]\[ \left[\begin{array}{l}g \\ h\end{array}\right] = \left[\begin{array}{rr}10 & -6 \\ -5 & 8\end{array}\right]\left[\begin{array}{l}62 \\ 70\end{array}\right] \][/tex]
Option C:
[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]
This would imply transformed coefficients (0.2, -0.12, -0.1, 0.16) being those required for the solution.
Option D:
[tex]\[ \left[\begin{array}{l}9 \\ h\end{array}\right] = \left[\begin{array}{cc}8 & 6 \\ 5 & 10\end{array}\right]\left[\begin{array}{l}62 \\ 70\end{array}\right] \][/tex]
This option erroneously uses '9' instead of 'g' on the left-hand side and is likely incorrect syntactically and systematically in solving.
Finally, considering numerical validation from correct matrix inverses and checks:
The correct answer, as inferred from proper matrix inversion and solution application, is:
[tex]\[ C \][/tex]
So, the desired equation matching this configuration is:
Option C:
\[ \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]
