This question was not written properly
A Broadway theater has 600 seats, divided into orchestra, main, and balcony seating. Orchestra seats sell for $50, main seats for $35, and balcony seats for $25. If all the seats are sold, the gross revenue to the theater is $20,700. If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $17,700. How many are there of each kind of seat?
Answer:
Orchestra = X = 270 seats
Main = Y = 350 seats
Balcony = Z
Step-by-step explanation:
Let's represent number of sits in:
Orchestra = X
Main = Y
Balcony = Z
A Broadway theater has 600 seats, divided into orchestra, main, and balcony seating.
X + Y + Z = 600
X = 600 - Y - Z
Orchestra seats sell for $50, main seats for $35, and balcony seats for $25. If all the seats are sold, the gross revenue to the theater is $20,700.
Hence,
50× X + 35 × Y + $25 × Z = $20,700
50X + 35Y + 25Z = 20,700....... Equation 1
If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $17,700.
50×1/2(X) + 35Y + 35Z = $17,700
25X + 35Y + 25Z = $17,700........ Equation 2
600 - Y - Z for X in Equation 1 and 2
Equation 1
50(600 - Y - Z) + 35Y + 25Z = 20,700
30,000 - 50Y - 50Z + 35Y + 25Z = 20,700
Collect like terms
30,000 - 20,700 = 50Y + 50Z - 35Y - 25Z
9,300 = 15Y + 25Z ......... Equation 3
From Equation 2
25(600 - Y - Z) + 35Y + 25Z= $17,700
15,000 -25Y - 25Z + 35Y + 25Z = 17,700
35Y - 25Y + 25Z - 25Z = 17,700 - 15,000
10Y + 0Z = 2,700.......
10Y = 2,700
Y = 2700/10
Y = 270
Hence:
15Y + 15Z = 9,300......... Equation 3
15(270) + 15Z = 9,300
4050 + 15Z = 9,300
15Z = 9300 - 4050
15Z = 5250
Z = 5250/15
Z = 350
X + Y + Z = 600
270 + 350 + Z = 600
Z = 600 - 270 - 350
Z =
