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To solve the problem and fill in the table, we need to allocate production hours at plants A and B based on the orders for car frames and truck frames.

### Given Information:
- **Plant A** produces:
  - 3 car frames per hour
  - 6 truck frames per hour
- **Plant B** produces:
  - 2 car frames per hour
  - 9 truck frames per hour

### Orders for each set:
1. **Order 1:**
   - Car frames: 654
   - Truck frames: 1518
2. **Order 2:**
   - Car frames: 393
   - Truck frames: 1611
3. **Order 3:**
   - Car frames: 835
   - Truck frames: 2355

### Formulation:
We need to determine how many hours each plant should work to produce the required frames.

Let:
- \( x_1 \), \( x_2 \), \( x_3 \) represent the hours Plant A works for Orders 1, 2, 3, respectively.
- \( y_1 \), \( y_2 \), \( y_3 \) represent the hours Plant B works for Orders 1, 2, 3, respectively.

The equations for each order are:

#### Order 1:
\[
\begin{align*}
3x_1 + 2y_1 &= 654 \quad (\text{car frames}) \\
6x_1 + 9y_1 &= 1518 \quad (\text{truck frames})
\end{align*}
\]

#### Order 2:
\[
\begin{align*}
3x_2 + 2y_2 &= 393 \quad (\text{car frames}) \\
6x_2 + 9y_2 &= 1611 \quad (\text{truck frames})
\end{align*}
\]

#### Order 3:
\[
\begin{align*}
3x_3 + 2y_3 &= 835 \quad (\text{car frames}) \\
6x_3 + 9y_3 &= 2355 \quad (\text{truck frames})
\end{align*}
\]

Let's solve these equations step by step to find the required hours. I'll proceed with the calculations.The number of hours each plant should operate to fulfill the orders is as follows:

### Order 1:
- **Plant A**: 190 hours
- **Plant B**: 42 hours

### Order 2:
- **Plant A**: 21 hours
- **Plant B**: 165 hours

### Order 3:
- **Plant A**: 187 hours
- **Plant B**: 137 hours

Let me know if you need any further details or clarifications!

Here are 5 related questions to extend your understanding:
1. How would the required hours change if Plant A's production rates were slightly higher?
2. What if Plant B could produce both car and truck frames at equal rates?
3. How would you handle an additional order with different car and truck frame requirements?
4. Can this problem be solved using matrix methods for more complex scenarios?
5. How would efficiency improvements at either plant affect the production hours?

**Tip**: When solving systems of equations involving production rates, it's useful to check for proportional relationships between different variables to simplify the process.

A supplier for the automobile industry manufactures car and truck frames at two different plants. How many hours should each plant be scheduled to operate to execute the orders in the right table? The production rates (in frames per hour) for each plant are given in the left table.

Explore how to allocate production hours at two different plants for car and truck frames using systems of linear equations. This detailed guide walks through the process using linear algebra concepts and Gaussian elimination to solve real-world industrial problems.

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