Select the correct answer.

Kyle and Lauren teach swimming lessons during the summer. Kyle has 8 classes with [tex]$g$[/tex] students in each class and 6 classes with [tex]$h$[/tex] students in each class. Lauren has 5 classes with [tex]$g$[/tex] students in each class and 10 classes with [tex]$h$[/tex] students in each class.

If Kyle has a total of 62 students and Lauren has a total of 70 students, which equation shows the solution to the system of equations that represents this situation?

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]

B. [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]

C. [tex]$\left[\begin{array}{l}g \\ h\end{array}\right]=\left[\begin{array}{cc}8 & 6 \\ 5 & 10\end{array}\right]\left[\begin{array}{l}62 \\ 2 \\ 70\end{array}\right]$[/tex]

D. [tex]$\left[\begin{array}{l}g \\ h\end{array}\right]=\left[\begin{array}{cc}10 & -6 \\ -5 & 8\end{array}\right]\left[\begin{array}{l}62 \\ 70\end{array}\right]$[/tex]



Answer :

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]