To determine Suzie's hourly wages for her jobs in tutoring, babysitting, and at the grocery store, we can set up a matrix equation based on the data provided.
The matrix equation we want to solve is of the form [tex]\(Ax = B\)[/tex], where:
- [tex]\(A\)[/tex] is a matrix representing the hours worked each week at each job.
- [tex]\(x\)[/tex] is a vector containing the hourly wages for each job.
- [tex]\(B\)[/tex] is a vector representing the total earnings each week.
Given the details:
- In Week 1, Suzie worked 4 hours tutoring, 8 hours babysitting, and 6 hours at the grocery store.
- In Week 2, she worked 3 hours tutoring, 5 hours babysitting, and 4 hours at the grocery store.
- In Week 3, she worked 4 hours tutoring, 6 hours babysitting, and 4 hours at the grocery store.
And she earned:
- \[tex]$215 in Week 1
- \$[/tex]145 in Week 2
- \[tex]$170 in Week 3
We can form the following equation:
\[
\begin{bmatrix}
4 & 8 & 6 \\
3 & 5 & 4 \\
4 & 6 & 4
\end{bmatrix}
\begin{bmatrix}
t \\
b \\
g
\end{bmatrix} =
\begin{bmatrix}
215 \\
145 \\
170
\end{bmatrix}
\]
Here:
- \( \begin{bmatrix} 4 & 8 & 6 \\ 3 & 5 & 4 \\ 4 & 6 & 4 \end{bmatrix} \) is the matrix \(A\) representing the hours worked each week.
- \( \begin{bmatrix} t \\ b \\ g \end{bmatrix} \) is the vector \(x\) representing the hourly wages for tutoring (\(t\)), babysitting (\(b\)), and the grocery store (\(g\)).
- \( \begin{bmatrix} 215 \\ 145 \\ 170 \end{bmatrix} \) is the vector \(B\) representing the total earnings each week.
When we solve this system of linear equations, we find that the hourly wages are:
\[ t = 15.00 \]
\[ b = 10.00 \]
\[ g = 12.50 \]
Therefore, Suzie's hourly wages are:
- \$[/tex]15.00 per hour for tutoring.
- \[tex]$10.00 per hour for babysitting.
- \$[/tex]12.50 per hour for the grocery store.
