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Let's break down the information given to form equations for this problem:

1. Let $ x $, $ y $, and $ z $ represent the units of products A, B, and C, respectively.

2. **Total units produced constraint**:  
   \[
   x + y + z = 9000
   \]

3. **Profit constraint**:  
   - Profit per unit for A, B, and C is $1, $2, and $3, respectively.
   - Total profit = $20,000.
   \[
   1x + 2y + 3z = 20000
   \]

4. **Cost constraint**:  
   - Fixed costs are $16,000.
   - Production costs per unit for A, B, and C are $4, $5, and $7, respectively.
   - Total cost (including fixed cost) = $68,000.
   \[
   4x + 5y + 7z + 16000 = 68000
   \]
   Simplifying, we get:
   \[
   4x + 5y + 7z = 52000
   \]

We now have a system of three equations:
\[
\begin{cases}
x + y + z = 9000 \\
x + 2y + 3z = 20000 \\
4x + 5y + 7z = 52000
\end{cases}
\]

Let's solve this system of equations to find $ x $, $ y $, and $ z $.The solution to the system of equations is:

- $ x = 3000 $ units of product A,
- $ y = 1000 $ units of product B,
- $ z = 5000 $ units of product C.

Therefore, the manufacturer should produce:
- 3000 units of product A,
- 1000 units of product B,
- 5000 units of product C.

Would you like a detailed explanation of the steps taken, or do you have any questions?

### Related Questions
1. How would the solution change if the profit per unit of product A increased?
2. What if the production cost per unit of product C was reduced to $6?
3. How would the equations change if the fixed cost increased to $18,000?
4. What is the impact on production numbers if the profit goal was raised to $25,000?
5. Could these constraints lead to an optimal solution using linear programming?

### Tip
In problems like this, setting up clear equations from the constraints is key to solving systems accurately.

A manufacturer produces three products, A, B, and C. The profits for each unit of A, B, and C sold are $1, $2, and $3, respectively. Fixed costs are $16,000 per year, and the costs of producing each unit of A, B, and C are $4, $5, and $7, respectively. Next year, a total of 9000 units of all three products is to be produced and sold, and a total profit of $20,000 is to be realized. If total cost is to be $68,000, how many units of each of the products should be produced next year?

Learn how to determine the optimal production numbers for products A, B, and C, maximizing profit while adhering to cost constraints. This problem involves algebra and systems of linear equations, ideal for high school students in Grades 10-12.

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