A Broadway theater has 500 seats divided into orchestra main

A Broadway theater has 500 seats, divided into orchestra, main, and balcony seating. Orchestra seats sell for dollar-sign 50 comma main seats for dollar-sign 35 comma and balcony seats for dollar-sign 25 point If all the seats are sold, the gross revenue to the theater is dollar-sign 17 comma 100 point If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is dollar-sign 14 comma 600 point How many are there of each kind of seat?

Solution

let the numbe of orchestra seats be = x
let the numbe of main seats be = y
let the numbe of balcony seats be = z

each orchestra seat is sold for = $ 50

each main seat is sold for = $ 35

each bacony seat is sold for = $ 50

total seats = 500

=> x+y+z= 500 , ---------(1)

when all the seats are sold then gross revenue is = $ 17100

=> 50x+35y+25z = 17100 , ---------(2)

when all the main and balcony seats are sold and half of the orchestra seats are sold then the gross revenue is = $ 14600

=> 50(x/2)+35y+25z = 14600

or 25x+35y+25z = 14600 , -----------(3)

so now we have three equation in three variables

using substitution method

subtract (3) from (2)
25x=2500
x= 2500/25
x= 100 , --------------> orchestra seats

multiply (1) by 25
25*100+25y+25z=12500

25y+25z=12500-2500
25y+25z=10000 , ----------- (4)

equation (2) = 5000+35y+25z=17100
35y+25z=12100 , ------------- (5)

subtract (4) from (5)
10y=2100
y=210 , ------------> main seats

we could find z form the equation (1)   
x+y+z=500

100+210+z=500

=> z= 190, -----------> balcony seats