Math  /  Algebra

QuestionA 5000-seat theater has tickets for sale at $27\$ 27 and $40\$ 40. How many tickets should be sold at each price for a sellout performance to generate a total revenue of $149,300\$ 149,300 ?
The number of tickets for sale at $27\$ 27 should be \square

Studdy Solution

STEP 1

What is this asking? If all 5000 seats are sold, how many should be $27\$27 tickets and how many should be $40\$40 tickets to make a total of $149,300\$149,300? Watch out! Don't mix up the number of tickets with the price of the tickets!

STEP 2

1. Set up variables
2. Create equations
3. Solve the system of equations

STEP 3

Let's call the number of $27\$27 tickets "xx".
And the number of $40\$40 tickets will be "yy".
Easy peasy!

STEP 4

We know the theater has **5000** seats, and if it sells out, that means x+y=5000x + y = 5000.
This is our **first equation**!

STEP 5

We also know the total revenue should be $149,300\$149,300.
Since each xx ticket costs $27\$27 and each yy ticket costs $40\$40, we can write this as 27x+40y=14930027 \cdot x + 40 \cdot y = 149300.
This is our **second equation**!

STEP 6

We can rewrite the first equation to be y=5000xy = 5000 - x.
This helps us to express yy in terms of xx.

STEP 7

Now, let's plug this value of yy into our second equation: 27x+40(5000x)=14930027 \cdot x + 40 \cdot (5000 - x) = 149300.

STEP 8

Distribute the 40: 27x+20000040x=14930027 \cdot x + 200000 - 40 \cdot x = 149300. Combine the xx terms: 13x+200000=149300-13 \cdot x + 200000 = 149300. Subtract 200000 from both sides: 13x=149300200000-13 \cdot x = 149300 - 200000. Simplify: 13x=50700-13 \cdot x = -50700. Divide both sides by -13: x=5070013x = \frac{-50700}{-13}. So, x=3900x = 3900.
This means we need to sell **3900** tickets at $27\$27!

STEP 9

Now, plug x=3900x = 3900 back into the equation y=5000xy = 5000 - x. So, y=50003900y = 5000 - 3900. Therefore, y=1100y = 1100.
This means we need to sell **1100** tickets at $40\$40!

STEP 10

We should sell **3900** tickets at $27\$27 and **1100** tickets at $40\$40.

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