John, Micheal and Christopher are running a supermarket. To deliver the products they use three different types of carry bags, they are polythene bags, handmade bags and newspaper envelopes. It is found that John, Michael, and Christopher are using 20, 30 and 40 polythene bags. 30, 40 and 20 handmade bags, and finally 40, 20 and 30 newspaper envelopes respectively in a month. The expenditures for procuring these bags are 250, 270 and 200. Use the above situation to answer 1. Cost of each bag 2. Which shopkeeper is best financially? 3. Which shopkeeper is best environmentally?                                                                                                                      Solve this using cramer's rule

To solve this problem using Cramer's Rule, we will follow these steps:

### 1. Set Up the System of Equations
The system of equations based on the problem is:

For John:
\[
20x + 30y + 40z = 250
\]

For Michael:
\[
30x + 40y + 20z = 270
\]

For Christopher:
\[
40x + 20y + 30z = 200
\]

### 2. Represent the Coefficients as a Matrix
Let the coefficient matrix $ A $ be:
\[
A = \begin{pmatrix}
20 & 30 & 40 \\
30 & 40 & 20 \\
40 & 20 & 30
\end{pmatrix}
\]

Let the constant matrix $ B $ be:
\[
B = \begin{pmatrix}
250 \\
270 \\
200
\end{pmatrix}
\]

The system of equations can be represented as:
\[
A \cdot \begin{pmatrix} x \\ y \\ z \end{pmatrix} = B
\]

### 3. Cramer's Rule
Cramer's Rule states that the solution for each variable is given by:
\[
x = \frac{\text{det}(A_x)}{\text{det}(A)}, \quad y = \frac{\text{det}(A_y)}{\text{det}(A)}, \quad z = \frac{\text{det}(A_z)}{\text{det}(A)}
\]
Where:
- $ A_x $ is the matrix formed by replacing the first column of $ A $ with $ B $.
- $ A_y $ is the matrix formed by replacing the second column of $ A $ with $ B $.
- $ A_z $ is the matrix formed by replacing the third column of $ A $ with $ B $.

### 4. Calculate Determinants

Let's calculate the determinants to find the costs of each bag.Using Cramer's Rule, the costs of each type of bag are calculated as follows:

- Cost of one polythene bag ($x$) = \$1
- Cost of one handmade bag ($y$) = \$5
- Cost of one newspaper envelope ($z$) = \$2

### 2. Which Shopkeeper is Best Financially?
Now, let's determine which shopkeeper is best financially by calculating their total expenditures:

- **John's Expenditure**:  
  $ 20 \times 1 + 30 \times 5 + 40 \times 2 = 20 + 150 + 80 = 250 $
  
- **Michael's Expenditure**:  
  $ 30 \times 1 + 40 \times 5 + 20 \times 2 = 30 + 200 + 40 = 270 $
  
- **Christopher's Expenditure**:  
  $ 40 \times 1 + 20 \times 5 + 30 \times 2 = 40 + 100 + 60 = 200 $

**Christopher** has the lowest expenditure, making him the best financially.

### 3. Which Shopkeeper is Best Environmentally?
From an environmental perspective:
- Polythene bags are the least eco-friendly.
- Handmade bags are more eco-friendly.
- Newspaper envelopes are the most eco-friendly.

Evaluating their usage:
- **John**: Uses 20 polythene bags, 30 handmade bags, and 40 newspaper envelopes.
- **Michael**: Uses 30 polythene bags, 40 handmade bags, and 20 newspaper envelopes.
- **Christopher**: Uses 40 polythene bags, 20 handmade bags, and 30 newspaper envelopes.

**John** uses the most newspaper envelopes and the fewest polythene bags, so he is the best environmentally.

---

Would you like further details on any part of this solution? Here are some related questions:

1. How do you find the determinant of a 3x3 matrix?
2. What other methods can solve a system of linear equations?
3. How does Cramer's Rule work for larger systems?
4. What are the pros and cons of using polythene bags versus newspaper envelopes?
5. How can businesses reduce their environmental impact by changing the materials they use?

**Tip**: Cramer's Rule is a useful method for solving small systems of linear equations, but it can be computationally expensive for larger systems.

Learn how to solve the costs of polythene bags, handmade bags, and newspaper envelopes for John, Michael, and Christopher using Cramer's Rule. Understand the financial implications and environmental impacts based on their bag usage.

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