Question 287564
A Broadway theater has 700 seats, divided into orchestra, main, and balcony seating. 
Orchestra seats sells for $40, Main for $30, and balcony for $20.
 If all seats are sold the gross revenue is $20,200.
 If all main and balcony seats are sold, but only half of the orchestra is sold,
 the gross revenue is $17,400.
 How many seats are there of each kind?
:
Let x = no. of orchestra seats
Let y = no. of main
Let z = no. of balcony
:
Total seat equation
x + y + z = 700
:
all seats revenue  equation
40x + 30y + 20z = 20200
:
half orch, all main, all balcony revenue equation
40(.5x) + 30y + 20z = 17400
20x + 30y + 20z = 17400
:
Use elimination on the last two equation to find x:
40x + 30y + 20z = 20200
20x + 30y + 20z = 17400
---------------------------subtraction eliminates y and z
20x = 2800
x = {{{2800/20}}}
x = 140 orchestra seats
:
Using the 1st equation, replace x with 140
140 + y + z = 700
y + z = 700 - 140
y + z = 560
y = (560-z)
:
Using substitution, findz
40x + 30y + 20z = 20200
40(140) + 30(560-z) + 20z = 20200
5600 + 16800 - 30z + 20z = 20200
22400 - 30z + 20z = 20200
-10z = 20200 - 22400
-10z = -2200
z = +220 balcony seat
:
I'll let you find y, the number of main seats
:
Orchestra-140
Main-
Balcony-220