Question

A Broadway theater has 300 total seats, divided into orchestra, main, and balcony seating. The total number of main and balcony seats combined is 3 times the number of orchestra seats. Orchestra seats sell for $60, main seats for $45, and balcony seats for $25. If all the seats are sold, the total revenue to the theater is $12,725. How many of each kind of seat are there? Set up and solve a system of equations. If you use matrices, list both the original matrix used and the reduced-row echelon (RREF) matrix

Answer 1

By setting up and solving a system of equations, we can find that the Broadway theater has 135 orchestra seats, 180 main seats, and 240 balcony seats.

Let x be the number of orchestra seats.

Let y be the number of main seats.

Let z be the number of balcony seats.

2. Translate the problem into equations:

Total seats: x + y + z = 300

Main and balcony combined: y + z = 3x (main and balcony are 3 times orchestra)

Revenue: 60x + 45y + 25z = 12725 (revenue from seat sales)

3. Solve the system:

Solve the second equation for y: y = 3x - z

Substitute this into the third equation: 60x + 45(3x-z) + 25z = 12725

Simplify and solve for z: 195x + 25z = 12725

z = 510 - 7.8x

Substitute z back into the first equation: x + 3x - (510 - 7.8x) = 300

x = 135

Finding y and z:

y = 3x - z = 180 and z = 510 - 7.8x = 240

Therefore, by setting up and solving a system of equations, we found that the Broadway theater has 135 orchestra seats, 180 main seats, and 240 balcony seats.