Question

select the correct answer.
molly is an insurance agent. she is mailing packages filled with marketing items to a few of her clients. the items she is including are pens that weigh x ounces, notepads that weigh y ounces, and magnets that weigh z ounces.

• a package filled with 3 pens, 2 notepads, and 1 magnet weighs 8 ounces.
• a package filled with 2 pens, 1 notepad, and 2 magnets weighs 3.9 ounces.
• a package filled with 4 pens and 3 magnets weighs 4 ounces.

which matrix equation can be solved to find the weight of each item?
a.
\

$$\begin{bmatrix}3 & 2 & 1\\\\2 & 1 & 2\\\\4 & 0 & 3\\end{bmatrix}$$

\

$$\begin{bmatrix}x\\\\y\\\\z\\end{bmatrix}$$

=\

$$\begin{bmatrix}8\\\\3.9\\\\4\\end{bmatrix}$$

b.
\

$$\begin{bmatrix}3 & 2 & 1\\\\2 & 1 & 2\\\\1 & 2 & 3\\end{bmatrix}$$

\

$$\begin{bmatrix}x\\\\y\\\\z\\end{bmatrix}$$

=\

$$\begin{bmatrix}8\\\\3.9\\\\4\\end{bmatrix}$$

c.
\

$$\begin{bmatrix}3 & 2 & 1\\\\2 & 1 & 3\\\\1 & 2 & 0\\end{bmatrix}$$

\

$$\begin{bmatrix}x\\\\y\\\\z\\end{bmatrix}$$

=\

$$\begin{bmatrix}8\\\\3.9\\\\4\\end{bmatrix}$$

d.
\

$$\begin{bmatrix}3 & 2 & 1\\\\2 & 1 & 2\\\\4 & 3 & 0\\end{bmatrix}$$

\

$$\begin{bmatrix}x\\\\y\\\\z\\end{bmatrix}$$

=\

$$\begin{bmatrix}8\\\\3.9\\\\4\\end{bmatrix}$$

Explanation:

Step1: Define Variables and Equations

Let \( x \) = weight of pens, \( y \) = weight of notepads, \( z \) = weight of magnets.
- Package 1: \( 3x + 2y + z = 8 \)
- Package 2: \( 2x + y + 2z = 3.9 \)
- Package 3: \( 4x + 3y + 0z = 4 \) (Wait, no—wait, the third package is 4 pens, 3 magnets? Wait, re-reading:
Wait, the three packages:
1. 3 pens, 2 notepads, 1 magnet: \( 3x + 2y + z = 8 \)
2. 2 pens, 1 notepad, 2 magnets: \( 2x + y + 2z = 3.9 \)
3. 4 pens, 0 notepads, 3 magnets: Wait, no, the third package: "A package filled with 4 pens and 3 magnets weighs 4 ounces." So \( 4x + 0y + 3z = 4 \)? Wait, no, the options: Let's check the coefficient matrices.

Looking at the options, the coefficient matrix (left) should have rows as packages, columns as \( x, y, z \).

Package 1: 3 pens, 2 notepads, 1 magnet → row: [3, 2, 1]
Package 2: 2 pens, 1 notepad, 2 magnets → row: [2, 1, 2]
Package 3: 4 pens, 0 notepads, 3 magnets? Wait, no, the third package: "4 pens and 3 magnets" → so \( y = 0 \), so row: [4, 0, 3]? Wait, no, the options:

Wait option A:
Coefficient matrix:
\[ \begin{bmatrix} 3 & 2 & 1 \\ 2 & 1 & 2 \\ 4 & 0 & 3 \\ \end{bmatrix} \]
Constants: \( \begin{bmatrix} 8 \\ 3.9 \\ 4 \end{bmatrix} \)

Wait no, the second package: "A package filled with 2 pens, 1 notepad, and 2 magnets weighs 3.9 ounces." So \( 2x + y + 2z = 3.9 \)
Third package: "A package filled with 4 pens and 3 magnets weighs 4 ounces." So \( 4x + 0y + 3z = 4 \)

Wait but option A's third row is [4, 0, 3]? Wait no, option A's third row is [4, 0, 3]? Wait the first option (A) has:

First row: 3, 2, 1
Second row: 2, 1, 2
Third row: 4, 0, 3
Constants: 8, 3.9, 4.

Wait but let's check the second package again: "2 pens, 1 notepad, 2 magnets" → \( 2x + y + 2z = 3.9 \) (matches second row [2,1,2])
First package: 3 pens, 2 notepads, 1 magnet → \( 3x + 2y + z = 8 \) (matches first row [3,2,1])
Third package: 4 pens, 0 notepads, 3 magnets → \( 4x + 0y + 3z = 4 \) (matches third row [4,0,3] in option A? Wait no, option A's third row is [4,0,3]? Wait the image:

Wait the user's image:

Option A:
\[ \begin{bmatrix} 3 & 2 & 1 \\ 2 & 1 & 2 \\ 4 & 0 & 3 \\ \end{bmatrix} \]
times \( \begin{bmatrix} x \\ y \\ z \end{bmatrix} \) equals \( \begin{bmatrix} 8 \\ 3.9 \\ 4 \end{bmatrix} \)

Wait but the third package is "4 pens and 3 magnets" → so 4x + 3z = 4 (since no notepads, y=0). So the third row should be [4, 0, 3], which is option A's third row.

Wait but let's check the second package: "2 pens, 1 notepad, 2 magnets" → row [2,1,2], which is option A's second row.

First package: [3,2,1], option A's first row.

So the correct matrix equation is option A? Wait no, wait the second package's weight is 3.9, first is 8, third is 4. So the constants are [8, 3.9, 4], and the coefficient matrix is as in option A. Wait but let's check the other options.

Option B:
Coefficient matrix:
\[ \begin{bmatrix} 3 & 2 & 4 \\ 2 & 1 & 2 \\ 1 & 2 & 3 \\ \end{bmatrix} \]
No, that doesn't match.

Option C:
\[ \begin{bmatrix} 3 & 2 & 4 \\ 2 & 1 & 3 \\ 1 & 2 & 0 \\ \end{bmatrix} \]
No.

Option D:
\[ \begin{bmatrix} 3 & 2 & 1 \\ 2 & 1 & 2 \\ 4 & 3 & 0 \\ \end{bmatrix} \]
Third row: 4,3,0 → which would be 4 pens, 3 notepads, 0 magnets, but the third package has 0 notepads. So no.

Wait, I think I made a mistake. Wait the third package: "A package filled with 4 pens and 3 magnets" → so notepads (y) = 0, so the third row is [4, 0, 3]. So the coefficient matrix should have third row [4, 0, 3], which is option A. So the correct matrix equation is option A.

Step2: Verify Coefficients

- Package 1: 3x + 2y + 1z = 8 ✔️
- Package 2: 2x + 1y + 2z = 3.9 ✔️
- Package 3: 4x + 0y + 3z = 4 ✔️

Thus, the correct matrix is option A.

Answer:

A. \(

$$\begin{bmatrix} 3 & 2 & 1 \\ 2 & 1 & 2 \\ 4 & 0 & 3 \end{bmatrix}$$

$$\begin{bmatrix} x \\ y \\ z \end{bmatrix}$$

=

$$\begin{bmatrix} 8 \\ 3.9 \\ 4 \end{bmatrix}$$

\)