A 6000-seat theater has tickets for sale at $28 and $40How many tickets should be sold at each price for a sellout performance to generate a total revenue of $190,800

Let's define the variables:

- Let $ x $ be the number of tickets sold at $28.
- Let $ y $ be the number of tickets sold at $40.

We know the following:
1. The total number of seats is 6000, so:
   \[
   x + y = 6000
   \]
2. The total revenue from the sellout is $190,800, so:
   \[
   28x + 40y = 190,800
   \]

Now we have a system of two equations:

1. $ x + y = 6000 $
2. $ 28x + 40y = 190,800 $

### Solving the system of equations:

From the first equation, solve for $ y $:
\[
y = 6000 - x
\]

Substitute this expression for $ y $ into the second equation:
\[
28x + 40(6000 - x) = 190,800
\]

Simplify the equation:
\[
28x + 240,000 - 40x = 190,800
\]
\[
-12x + 240,000 = 190,800
\]
\[
-12x = 190,800 - 240,000
\]
\[
-12x = -49,200
\]
\[
x = \frac{-49,200}{-12} = 4100
\]

Now that we have $ x = 4100 $, substitute this back into the first equation to find $ y $:
\[
4100 + y = 6000
\]
\[
y = 6000 - 4100 = 1900
\]

### Conclusion:
- 4100 tickets should be sold at $28.
- 1900 tickets should be sold at $40.

Would you like a breakdown of these steps or have any specific questions?

Here are 5 related questions to explore further:
1. What would happen if the ticket prices changed to $30 and $45?
2. How would the solution change if the total revenue were $200,000?
3. Can this type of problem be solved using matrix methods?
4. What if the theater sold tickets at three different price points?
5. How does changing the number of seats impact the ticket distribution?

**Tip**: Always re-check your substitution to ensure accuracy in solving systems of equations.

A 6000-seat theater has tickets for sale at $28 and $40. How many tickets should be sold at each price for a sellout performance to generate a total revenue of $190,800?

Learn how to solve a system of equations to determine how many tickets should be sold at $28 and $40 in a 6000-seat theater to achieve a total revenue of $190,800. This step-by-step guide covers key algebraic methods, including the substitution and elimination methods. Suitable for students in Grades 9-11.

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