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Question 1198615: The ABC Ink Company is a small family owned company that sells packages of ink cartridge refills
for smartpens. The Xavier set contains one blue ink refill and one black ink refill. The Yvonne set
includes two blue ink refills, three black ink refills, and one red ink refill. The Zena set includes
four blue ink refills, five black ink refills, and one red ink refill. The company has sold most of its
stock and has found that it has only 11 blue ink cartridge refills, 14 black ink cartridge refills, and 3
red ink cartridge refills. How many of each set should the company package to sell in order to use
all of the remaining ink cartridges so that there will be none left in inventory.
a. Describe what the variables represent: =
x=
y=
z=
b. Write the system of linear equations:
c. Solve the system of equations using Gauss-Jordan Elimination. Show all proper row operations
and the resulting matrices. Write the solution as an ordered triple, if appropriate.
d. Fill out the table to show the possible combinations of sets the company can package:
x y z
e. In at least one complete sentence with proper grammar and correct spelling, write the
solutions in terms of what the variables represent.
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website! **a. Define Variables:**
* **x:** Number of Xavier sets to be packaged.
* **y:** Number of Yvonne sets to be packaged.
* **z:** Number of Zena sets to be packaged.
**b. System of Linear Equations:**
* **Blue Ink Refills:** 1x + 2y + 4z = 11
* **Black Ink Refills:** 1x + 3y + 5z = 14
* **Red Ink Refills:** 0x + 1y + 1z = 3
**c. Gauss-Jordan Elimination**
1. **Augmented Matrix:**
```
[ 1 2 4 | 11 ]
[ 1 3 5 | 14 ]
[ 0 1 1 | 3 ]
```
2. **Row Operations:**
* **R2 = R2 - R1:**
```
[ 1 2 4 | 11 ]
[ 0 1 1 | 3 ]
[ 0 1 1 | 3 ]
```
* **R3 = R3 - R2:**
```
[ 1 2 4 | 11 ]
[ 0 1 1 | 3 ]
[ 0 0 0 | 0 ]
```
* **R1 = R1 - 2R2:**
```
[ 1 0 2 | 5 ]
[ 0 1 1 | 3 ]
[ 0 0 0 | 0 ]
```
3. **Reduced Row-Echelon Form:**
```
[ 1 0 2 | 5 ]
[ 0 1 1 | 3 ]
[ 0 0 0 | 0 ]
```
**d. Possible Combinations of Sets**
* From the reduced row-echelon form:
* x + 2z = 5
* y + z = 3
* **Possible Solutions:**
* Let z = 0:
* x = 5, y = 3
* Let z = 1:
* x = 3, y = 2
* Let z = 2:
* x = 1, y = 1
* **Table of Possible Combinations:**
| x (Xavier) | y (Yvonne) | z (Zena) |
|---|---|---|
| 5 | 3 | 0 |
| 3 | 2 | 1 |
| 1 | 1 | 2 |
**e. Solution Interpretation:**
The company can package the remaining ink cartridges in three possible ways:
* 5 Xavier sets, 3 Yvonne sets, and 0 Zena sets.
* 3 Xavier sets, 2 Yvonne sets, and 1 Zena set.
* 1 Xavier set, 1 Yvonne set, and 2 Zena sets.
This will ensure all 11 blue, 14 black, and 3 red ink cartridges are used.
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