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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?
Orchestra-
Main-
Balcony-
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! 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 = 
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
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