5.1 Solve Systems of \( \leftarrow \quad \) A 5000 -seat theater has tickets for sale at \( \$ 27 \) and \( \$ 40 \). How many tickets should be sold at each price for a sellout performance to generate a total revenue of \( \$ 151,900 \) ?

To solve this, let \( x \) be the number of $27 tickets sold and \( y \) be the number of $40 tickets sold. You have two equations based on the information given: 1. The total number of tickets equation: \( x + y = 5000 \) 2. The total revenue equation: \( 27x + 40y = 151900 \) Now, you can solve this system of equations. First, rearrange the first equation to find \( y \): \( y = 5000 - x \) Substituting into the revenue equation: \( 27x + 40(5000 - x) = 151900 \) This simplifies to: \( 27x + 200000 - 40x = 151900 \) Combine like terms: \( -13x + 200000 = 151900 \) Now, isolate \( x \): \( -13x = 151900 - 200000 \) \( -13x = -48000 \) Thus, \( x = \frac{48000}{13} \approx 3692 \) (approximately) Now using \( y = 5000 - x \): \( y = 5000 - 3692 = 1308 \) So, to achieve a sellout performance while hitting the revenue target, you should sell approximately 3692 tickets at $27 and 1308 tickets at $40. Cheers to a profitable night at the theatre!