A Broadway theater has 500500 seats, divided into orchestra, main, and balcony seating. Orchestra seats sell for $ 60 comma$60, main seats for $ 45 comma$45, and balcony seats for $ 30.$30. If all the seats are sold, the gross revenue to the theater is $ 21 comma 600.$21,600. If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $ 18 comma 600.$18,600. How many are there of each kind of seat?

The number of orchestra seats=100

The number of main seats=240

The number of balcony seats=160

Step-by-step explanation:

Total number of seats in the theater=500

Let the number of orchestra seats=x

Let the number of main seats=y

Let the number of balcony seats=z

x+y+z=500

Cost of Orchestra Seats =$60

Cost of Main Seats =$45

Cost of Balcony Seats =$30

If all the seats are sold, the gross revenue to the theater is $ 21,600.

60x+45y+30z=21,600.

If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $18,600.

45y+30z+x(60)=18,600

We solve the three equations simultaneously

x+y+z=500

60x+45y+30z=21,600.

30x+45y+30z=18,600

Subtract the third equation from the second

30x=3000

x=100

Substitute x=100 into the equations x+y+z=500

100+y+z=500

y+z=500-100

y=400-z

Substitute y=400-z and x=100 into 30x+45y+30z=18,600

30x+45y+30z=18,600

30(100)+45(400-z)+30z=18600

3000+18000-45z+30z=18600

-15z=18600-21000

-15z=-3600

z=240

y=400-z=400-240=160

Therefore: x=100, y=240, z=160

The number of orchestra seats=100

The number of main seats=240

The number of balcony seats=160

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