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]
