system of equations

Question

The local theater has three types of seats for Broadway plays: main floor, balcony, and mezzanine. Main floor tickets are $40$ balcony tickets are $32, and mezzanine tickets are $28 One particular night, sales totaled $31,760 There were 385 more main floor tickets sold than balcony and mezzanine tickets combined. The number of balcony tickets sold is 30 more than 3times the number of mezzanine tickets sold. How many of each type of ticket were sold?

Answer 1

Let "b" represent the number of balcony tickets, "m" the number of mezzanine tickets, and "f" the number of main floor tickets.

The value of tickets sold then would be 40f + 32b + 28m and we are told that equals 31760 so the first equation is:

40f + 32b + 28m = 31760 or, dividing both sides by 4 we get

1) 10f + 8b + 7m = 7940

We are also told that f - 385 = b + m so equation #2 is:

2) f - b - m = 385

And we are told that b = 3m + 30 so the third equation is:

3) b - 3m = 30

Working with equations 1 & 2, if we can eliminate "f", it will have variables "b" and "m" just like equation #3. To eliminate the "f" variable, we can multiply equation #2 by -10 and add the result to equation #1 giving us a new equation #4:

1) 10f + 8b + 7m = 7940

2 modified) -10f +10b + 10m = -3850

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4) 18b + 17m = 4090

If we multiply equation #3 by -18 and add it to equation #4 we get:

3 modified) -18b + 54m = - 540

4) 18b + 17m = 4090

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71m = 3550

m = 50

So the number of mezzanine tickets sold was 50. Using Equation #3 (b - 3m = 30) we can plug in m = 50 and solve for "b":

b - 3(50) = 30

b - 150 = 30

b = 180

So the number of balcony tickets sold was 180.

Using Equation #2 (f - b - m = 385) we can plug in m = 50 and b = 180 to solve for "f":

f - 180 - 50 = 385

f - 230 = 385

f = 615

So there wer 615 floor tickets sold

Answer 2

Let's set up the system of equations. We can start with the part of the problem, "one particular night, sales totaled $31,760":

40F + 32B + 28M = 31760

where:

F is the number of main floor tickets

B is the number of balcony tickers

M is the number of mezzanine tickets

Next, let's set up the equation for, "there were 385 more main floor tickets sold than balcony and mezzanine tickets combined":

F = B + M + 385

Finally, let's look at the last equation for, "the number of balcony tickets sold is 30 more than 3 times the number of mezzanine tickets sold":

B = 30 + 3M

So now we have our three equations and 3 unknowns:

40F + 32B + 28M = 31760

F = B + M + 385

B = 30 + 3M

The last of these is the most simple, since it only has two variables. So let's substitute 30 + 3M for B in the second equation:

F = (30 + 3M) + M + 385

F = 4M + 415

And now we can substitute values for both B and F into the first equation to solve for M:

40(4M + 415) + 32(30 + 3M) + 28M = 31760

160M + 16600 + 960 + 96M + 28M = 31760

284M + 17560 = 31760

284M = 14200

M = 50

Now that we know M, we can solve for F:

F = 4M + 415

F = 4(50) + 415

F = 200 + 415

F = 615

And we can solve for B:

B = 30 + 3M

B = 30 + 3(50)

B = 30 + 150

B = 180

Now to check, we can plug all three of these numbers into the equations to see if they match:

40F + 32B + 28M = 31760

40(615) + 32(180) + 28(50) = 31760

24600 + 5760 + 1400 = 31760

31760 = 31760

F = B + M + 385

615 = 180 + 50 + 385

615 = 615

B = 30 + 3M

180 = 30 + 3(50)

180 = 30 + 150

180 = 180

All of the equations work (as expected), and now we can say with confidence that:

F = 615

B = 180

M = 50