Answer:
80 orchestra seats
180 main seats
140 balcony seats
Step-by-step explanation:
Total number of seats = 400
Orchestra seats sell for $80
Main seats sell for $65
Balcony seats sell for $40
If all the seats are sold, gross revenue = $23,700
If all the main seats and balcony seats are sold but only half of the orchestra seats are sold, the gross revenue = $20,500
Let x = Orchestra seats
Let y = main seats
Let z = balcony seats
x+y+z= 400............. (1)
Orchestra seats sell for $80, Main seats sell for $65 and bslcony seats sell for $40. The gross revenue if all the seats are sold = $23,700
80x + 65y + 40z = 23,700 ........(2)
If half of the orchestra seats are sold while all main and balcony seats are sold, we have
80(0.5x) + 65y+ 40z = 20,500
40x + 65y + 40z = 20,500 .....(3)
we have the simultaneous equations
x+y+z= 400............. (1)
80x + 65y + 40z = 23,700 ........(2)
40x + 65y + 40z = 20,500 .....(3)
Using elimination method to solve equation 2 and 3, we subtract equation 3 from 2
40x = 3200
x = 3200/40
x = 80 orchestra seats
Put x= 80 into equation 1 and 2
From equation 1,
x +y+z = 400
y + z = 400 - x
y+z = 400 -80
y+z = 320 .........(4)
From equation 2,
80x + 65y + 40z = 23,700
80(80) + 65y +40z = 23,700
6400 + 65y + 40z = 23,700
65y+40z = 23,700 - 6400
65y + 40z = 17,300 .........(5)
Multiply equation (4) by 40 and equation (5) by 1
So we have
40y + 40z = 12,800
65y + 40x = 17,300
subtract both equations from each other. we have
-25y = -4500
y = -4500/-25
y = 180 main seats
Put x = 80 and y= 180 into equation (1)
80+180+z = 400
260 + z = 400
z = 400-260
z = 140 balcony seats
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