Question

A theater charges ​$8 for​ main-floor seats and ​$4 for balcony seats. If all seats are​ sold, the ticket income is ​$3200. At one​ show, 25​% of the​ main-floor seats and 50​% of the balcony seats were sold and ticket income was ​$1000. How many seats are on the main floor and how many are in the​ balcony?

Answer 1

Let's use "m" for the number of main floor seats and "b" for the number of balcony seats.

One equation we have is based on selling out: 

     8m + 4b = 3200

The other is based on the other scenario:

   0.25•8m + 0.5•4b = 1000

which becomes:

            2m + 2b = 1000

So that's the system we need to solve:

     8m + 4b = 3200

     2m + 2b = 1000

You can use either substitution or elimination.  I'll show you elimination:

Take the second equation and multiply by -2 on both sides:

      8m + 4b = 3200
(-2)(2m + 2b) = -2(1000)

This gives us:

      8m + 4b = 3200
     -4m - 4b = -2000

Add the two equations and you get:

      4m = 1200

or      m = 300

Now solve for b:

          2m + 2b = 1000

     2(300) + 2b = 1000   (sub m = 300)

         600 + 2b = 1000

                    2b = 400     (subtract 600)

                      b = 200     (divide by 2)

So there are 300 main floor seats and 200 balcony seats.