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A shipping company charges different rates based on package weight:
- \[tex]$7 for packages weighing 5 pounds or less
- \$[/tex]15 for packages weighing more than 5 pounds but less than 10 pounds
- \[tex]$22 for packages weighing more than 10 pounds

During one hour, the company had 13 packages that totaled \$[/tex]168. The number of packages weighing 5 pounds or less was three more than those weighing more than 10 pounds.

The system of equations representing this situation is:
[tex]\[
\begin{array}{l}
s + m + l = 13 \\
7s + 15m + 22l = 168 \\
s - l = 3
\end{array}
\][/tex]

Which matrix can be used to show the number of packages in the different weight classes during that hour?

A. [tex]\(\left[\begin{array}{lll|l} 1 & 0 & 0 & 6 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 3 \end{array}\right]\)[/tex]

B. [tex]\(\left[\begin{array}{ccc|c} 1 & 0 & 0 & 10 \\ n & 1 & n & 3 \end{array}\right]\)[/tex]



Answer :

GinnyAnswer
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]

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