A Broadway theater has 800 seats, divided into orchestra, main, and balcony seating. Orchestra seats sell for $ 50, main seats for $ 40 , and balcony seats for $ 25. If all the seats are sold, the gross revenue to the theater is $ 30 comma 150. If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $ 26, 150. How many are there of each kind of seat?

Step-by-step explanation:

Let's represent the number of orchestra seats with the variable , the number of main seats with the variable , and the number of balcony seats with the variable .

From the first sentence, we know that the total number of seats shared among these three sections is 800, so we can setup the following equation:

From the second sentence, we can setup an equation for the total revenue by seat type as follows:

From the third sentence, we know that we cut the number of orchestra seats in half, and the revenue goes from $30150 to $26150, a difference of $4000. Since we know each orchestra seat is $50, we can divide into to determine there are orchestra seats, but remember, this number is HALF the total seats available, so there are total orchestra seats.

We can now plug in for in the two equations above:

Now we can solve one for in the first equation and plug it into the second equation:

We know know there are balcony seats. Finally, we can plug this number into our original equation to get the number of main seats:

So there are 160 orchestra seats, 410 main seats, and 230 balcony seats.

step-by-step explanation: