Enter the correct answer in the box.

Suzie tutors and babysits after school and works at a grocery store on the weekends. She earns a different hourly wage for each of the three jobs: [tex]$t$[/tex] dollars for tutoring, [tex]$b$[/tex] dollars for babysitting, and [tex][tex]$g$[/tex][/tex] dollars for working at the grocery store.

The table shows the number of hours she worked at each job over the course of three weeks.

\begin{tabular}{|l|c|c|c|}
\hline & Tutoring & Babysitting & \begin{tabular}{c}
Grocery \\
Store
\end{tabular} \\
\hline Week 1 & 4 & 8 & 6 \\
\hline Week 2 & 3 & 5 & 4 \\
\hline Week 3 & 4 & 6 & 4 \\
\hline
\end{tabular}

In week 1, Suzie earned [tex]$\$ 215$[/tex]; in week 2, she earned [tex]$\[tex]$ 145$[/tex][/tex]; and in week 3, she earned [tex]$\$ 170$[/tex].

What matrix equation could be used to determine the hourly wage she earns for each job? Fill in the missing elements in the equation.



Answer :

To determine the hourly wage Suzie earns for each job, we can set up a system of equations based on the information given. Let's represent the hourly wages for tutoring, babysitting, and working at the grocery store as [tex]\( t \)[/tex], [tex]\( b \)[/tex], and [tex]\( g \)[/tex] respectively.

From the problem statement, we have:
[tex]\[ \begin{align*} \text{Week 1:} & \quad 4t + 8b + 6g = 215 \\ \text{Week 2:} & \quad 3t + 5b + 4g = 145 \\ \text{Week 3:} & \quad 4t + 6b + 4g = 170 \end{align*} \][/tex]

These can be written in matrix form as:
[tex]\[ \begin{pmatrix} 4 & 8 & 6 \\ 3 & 5 & 4 \\ 4 & 6 & 4 \end{pmatrix} \begin{pmatrix} t \\ b \\ g \end{pmatrix} = \begin{pmatrix} 215 \\ 145 \\ 170 \end{pmatrix} \][/tex]

In this matrix equation, the matrix on the left-hand side represents the number of hours Suzie worked at each job each week, and the column vector represents her hourly wages. The right-hand side column vector represents the total earnings for each week.

Thus, the matrix equation is:
[tex]\[ \begin{pmatrix} 4 & 8 & 6 \\ 3 & 5 & 4 \\ 4 & 6 & 4 \end{pmatrix} \begin{pmatrix} t \\ b \\ g \end{pmatrix} = \begin{pmatrix} 215 \\ 145 \\ 170 \end{pmatrix} \][/tex]

Solving this system of linear equations will reveal the hourly wages [tex]\( t \)[/tex], [tex]\( b \)[/tex], and [tex]\( g \)[/tex]. The solution for these hourly wages is:
[tex]\[ t = 15.0, \quad b = 10.0, \quad g = 12.5 \][/tex]