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Let's analyze the problem and solve it step by step.

### Problem Summary:
The local theater sells three types of tickets:  
- Main floor tickets at \$45 each  
- Balcony tickets at \$42 each  
- Mezzanine tickets at \$37 each  

On one night:
- Total sales were \$95,411.
- The number of main floor tickets sold was 65 more than the combined number of balcony and mezzanine tickets.
- The number of balcony tickets sold was 208 more than four times the number of mezzanine tickets sold.

### Define Variables:
Let:
- $ x $ be the number of mezzanine tickets sold.
- $ y $ be the number of balcony tickets sold.
- $ z $ be the number of main floor tickets sold.

### Create Equations:
1. The total sales equation (based on the prices of each type of ticket):
   \[
   37x + 42y + 45z = 95,411
   \]
   
2. The relationship between balcony and mezzanine tickets:
   \[
   y = 4x + 208
   \]
   
3. The relationship between main floor and the sum of balcony and mezzanine tickets:
   \[
   z = x + y + 65
   \]

Now we can solve this system of equations. I'll proceed with the solution.The solution to the system of equations is as follows:

- The number of mezzanine tickets sold ($x$) is **173**.
- The number of balcony tickets sold ($y$) is **900**.
- The number of main floor tickets sold ($z$) is **1138**.

Would you like more details on the solution process or have any questions?

Here are five related questions to explore further:
1. How would the system change if the total sales were different?
2. Can we solve the system using matrix methods like Gaussian elimination?
3. How does changing the price of one type of ticket affect the total?
4. How could we model this problem using a graph?
5. How would adding a discount on ticket prices change the equations?

**Tip:** Breaking down word problems into small parts and defining variables helps simplify the solving process.

The local theater has three types of seats for Broadway plays: main floor ($45), balcony ($42), and mezzanine ($37). Total sales were $95,411. There were 65 more main floor tickets sold than the sum of balcony and mezzanine tickets. Balcony tickets sold were 208 more than four times the mezzanine tickets. How many of each ticket were sold?

Learn how to solve a system of equations involving Broadway ticket sales. This problem includes algebraic equations for calculating mezzanine, balcony, and main floor tickets sold. Total sales of $95,411, with price variations and relationships between ticket types.

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