The problem involves determining the number of packages in different weight classes that satisfy the given conditions.
To solve this, we developed a system of linear equations representing the situation. The three equations based on the given conditions are:
1. [tex]\( s + m + l = 13 \)[/tex] (The total number of packages is 13)
2. [tex]\( 7s + 15m + 22l = 168 \)[/tex] (The total cost for shipping these packages is \$168)
3. [tex]\( s - l = 3 \)[/tex] (The number of packages weighing five pounds or less is three more than those weighing more than ten pounds)
We represent this system of equations in an augmented matrix form:
[tex]\[
\left[\begin{array}{ccc|c}
1 & 1 & 1 & 13 \\
7 & 15 & 22 & 168 \\
1 & 0 & -1 & 3
\end{array}\right]
\][/tex]
This matrix precisely captures the system of equations laid out above, with each row representing one of the equations.
Therefore, the correct matrix to show the number of packages in the different weight classes during that hour is:
[tex]\[
\left[\begin{array}{ccc|c}
1 & 1 & 1 & 13 \\
7 & 15 & 22 & 168 \\
1 & 0 & -1 & 3
\end{array}\right]
\][/tex]
