Question

A Broadway theater has 700 ​seats, divided into​ orchestra, main, and balcony seating. Orchestra seats sell for $ 50 comma main seats for $ 40 comma and balcony seats for $ 25.  If all the seats are​ sold, the gross revenue to the theater is $ 26 comma 250.  If all the main and balcony seats are​ sold, but only half the orchestra seats are​ sold, the gross revenue is $ 22 comma 750. How many are there of each kind of​ seat?There are orchestra​ seats?

main​ seats?
balcony seats?

Answer 1

orchestra​ seats = 140 seats

main​ seats = 350 seats

balcony seats = 210 seats

Explanation:

Let x = number of orchestra seats

Let y = number of main seats

Let z = number of balcony seats

Theater has 700 seats. Thus;

Total seat equation is;

x + y + z = 700 - - - - (1)

We are told Orchestra seats sell for $ 50, main seats for $ 40, and balcony seats for $ 25 and a gross revenue of $26,250

Thus ;

all seats revenue equation is;

50x + 40y + 25z = 26250 - - - (eq2)

Now, we are told that, when all the main and balcony seats are​ sold, but only half the orchestra seats are​ sold, the gross revenue is $ 22,750

Thus, we have;

½(50x) + 40y + 25z = 22,750

Which gives ;

25x + 40y + 25z = 22,750 ---(eq3)

Solving the 3 equations simultaneously, we have;

x = 140

y = 350

z = 210