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?

Let's represent the number of orchestra seats with the variable [tex]o[/tex], the number of main seats with the variable [tex]m[/tex], and the number of balcony seats with the variable [tex]b[/tex].

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:

[tex]o + m + b = 800[/tex]

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

[tex]50o + 40m + 25b = 30150[/tex]

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 [tex]50[/tex] into [tex]4000[/tex] to determine there are [tex]80[/tex] orchestra seats, but remember, this number is HALF the total seats available, so there are [tex]160[/tex] total orchestra seats.

We can now plug in [tex]160[/tex] for [tex]o[/tex] in the two equations above:

[tex]160 + m + b = 800[/tex]

[tex]m + b = 640[/tex]

[tex]50(160) + 40m + 25b = 30150[/tex]

[tex]8000 + 40m + 25b = 22150[/tex]

Now we can solve one for [tex]m[/tex] in the first equation and plug it into the second equation:

[tex]m = 640 - b[/tex]

[tex]40(640 - b) + 25b = 22150[/tex]

[tex]25600 - 40b + 25b = 22150[/tex]

[tex]-15b = -3450[/tex]

[tex]b = 230[/tex]

We know know there are [tex]230[/tex] balcony seats. Finally, we can plug this number into our original equation to get the number of main seats:

[tex]m + 230 = 640[/tex]

[tex]m = 410[/tex]

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