A company makes two kinds of products: Basic and Advanced. To manufacture the Basic takes 2 hours in Assembly and 2.7 hours in Finishing. To manufacture the Advanced takes 2.6 hours in Assembly and 2.15 hours in Finishing. The Assembly department has 2,551 hours per week and the Finishing department has 2,496 hours per week. How many of each type of product should be manufactured to run at capacity?
If necessary, round down to the nearest whole number.
The number of Basic units should be 
 .
The number of Advanced units should be 
 .

To solve this problem, we need to set up a system of linear equations representing the time constraints for both the Assembly and Finishing departments. Let:

- \( x \) represent the number of Basic products.
- \( y \) represent the number of Advanced products.

The total time required for both products in the Assembly department is:
\[
2x + 2.6y = 2551
\]
The total time required for both products in the Finishing department is:
\[
2.7x + 2.15y = 2496
\]

We now solve this system of equations to determine \( x \) and \( y \).

First, we will use the method of substitution or elimination to solve these equations.The solution to the system of equations gives:

- \( x \approx 369.47 \) (Basic products)
- \( y \approx 696.95 \) (Advanced products)

Since we need to round down to the nearest whole number, the company should manufacture:

- 369 Basic units
- 696 Advanced units

Would you like more details on how these values were calculated, or do you have any further questions?

Here are five related questions for deeper understanding:
1. How does the total manufacturing time in each department affect the production numbers?
2. What happens if the available hours in the Assembly or Finishing departments change?
3. Could the company increase production by optimizing the manufacturing process?
4. What if the company introduces a third product with different time requirements?
5. How would an increase in demand affect production planning?

**Tip:** When solving systems of linear equations, substitution and elimination methods are powerful tools to handle real-world optimization problems like this one.

A company makes two kinds of products: Basic and Advanced. To manufacture the Basic takes 2 hours in Assembly and 2.7 hours in Finishing. To manufacture the Advanced takes 2.6 hours in Assembly and 2.15 hours in Finishing. The Assembly department has 2,551 hours per week and the Finishing department has 2,496 hours per week. How many of each type of product should be manufactured to run at capacity?

Learn how to solve a system of linear equations to optimize the production of Basic and Advanced products. This problem involves time constraints for Assembly and Finishing departments and applies key concepts of algebra and optimization. Suitable for high school students in Grades 10-12.

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