System of Equations | How would you solve this problem?

Question

A theater has 1,320 seats. Seats sell for $50 on the floor, $35 in the mezzanine, and $20 in the balcony. The number of seats on the balcony equals the total number of seats in the mezzanine and floor. Suppose the theater takes in $39,420 from each sold-out event. How many seats does the mezzanine section hold?

*Solve using substitution

Answer 1

Hi Bob. R

452 mezzanine seats

You can solve with 3 variables but they quickly reduce to two variables according to the second sentence

First you have totals

Total seats 1320

Total sales $39420

Then you have types of seat and their costs

Let

x = floor seats at $50 each

y = mezzanine seats at $35 each

z = balcony seats at $20 each

According to the second sentence

balcony seats = floor seats + mezzanine seats

z = x + y

So we can get rid of z with substitution

x + y + z = 1320 seats

Is also

x + y +(x + y) = 1320

2x + 2y = 1320

Moving just a little further

y = 660 -x

And for cost

50x + 35y + 20z = 39420

Is

50x + 35y + 20(x + y) = 39420

50x + 35y + 20x + 20y = 39420

70x + 55y = 39420

Then we can solve for x with substitution

70x + 55(660 -x) = 39420

Then go back and substitute in values to obtain both y and z where z = x + y

Give it a try

Answer 2

Let F be the number seats on the floor; M be the number of seats in the mezzanine; B be the number of seats in the balcony.

We know F+B+M = 1320 (since 1320 seats total)

We know B = M + F (look at second sentence)

We know 39420 = 50*F + 35*M + 20*B

You can solve the 3 by 3 system of equations

F+B+M =1320

B-M-F=0

50*F+35*M+20*B=39420

Please let me know if you need more help!