Question

Use a system of equations to solve the following problem.The local theater has three types of seats for Broadway plays: main floor, balcony, and mezzanine. Main floor tickets are$45, balcony tickets are$36, and mezzanine tickets are$32. One particular night, sales totaled$74,630. There were148 more main floor tickets sold than balcony and mezzanine tickets combined. The number of balcony tickets sold is429 more than2 times the number of mezzanine tickets sold. How many of each type of ticket were sold?

Answer 1

994 main floor tickets, 707 balcony tickets, 139 mezzanine tickets

Step-by-step explanation

The local theater has three types of seats:

Main floor = $45

Balcony tickets = $36

Mezzanine tickets = $32

Let the number main floor seats sold be x.

Let the number of Balcony tickets be y.

Let the number of Mezzanine tickets be z.

One night, all sales totalled $74,630.

This means that:

45 * x + 36 * y + 32 * z = 74,630

=> 45x + 36y + 32z = 74630 ______(1)

There were 148 more main floor tickets than balcony and mezzanine tickets combined, that is:

x = 148 + y + z _____(2)

The number of balcony tickets sold is 429 more than 2 times the number of mezzznine tickets sold. That is:

y = 429 + 2 * z

y = 429 + 2z _______(3)

Now, we have a system of three equations:

45x + 36y + 32z = 74630 _____(1)

x = 148 + y + z _____________(2)

y = 429 + 2z _______________(3)

Put (2) in (1):

45(148 + y + z) + 36y + 32z = 74630

6660 + 45y + 45z + 36y + 32z = 74630

Collect like terms:

45y + 36y + 45z + 32z = 74630 - 6660

81y + 77z = 67970_______(4)

Now, put (3) in (4):

81(429 + 2z) + 77z = 67970

34749 + 162z + 77z = 67970

Collect like terms:

239z = 67970 - 34749

239z = 33221

z = 33221 / 239

z = 139 tickets

Put the value of z in (3):

y = 429 + 2(139)

y = 278 + 429

y = 707 tickets

Put the values of y and z in (2):

x = 148 + 707 + 139

x = 994 tickets

Therefore, 994 main floor tickets, 707 balcony tickets, 139 mezzanine tickets were sold that night.