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 $59, balcony tickets are $50, and mezzanine tickets are $40. One particular night, sales totaled $73,785. There were 435 more main floor tickets sold than balcony and mezzanine tickets combined. The number of balcony tickets sold is 78 more than 3 times the number of mezzanine tickets sold. How many of each type of ticket were sold?
Why use a "system of equations", all the variables are easily defined .
number of mezzanines --- x
number of balconies ----- 3x + 78
number of main floor ---- x + 3x+78 + 435
= 4x + 513
59(4x+513) + 50(3x+78) + 40x = 73785
426x + 34167 = 73785
x = 93
back-sub into my definitions
To solve this problem, let's assign variables to the unknown quantities.
Let x represent the number of mezzanine tickets sold.
Let y represent the number of balcony tickets sold.
According to the problem, the number of main floor tickets sold is 435 more than the combined number of balcony and mezzanine tickets. Therefore, the number of main floor tickets sold can be expressed as (y + x) + 435.
The total sales revenue from each type of ticket can also be calculated. The revenue from mezzanine tickets is 40x, the revenue from balcony tickets is $50y, and the revenue from main floor tickets is $59 times the number of main floor tickets sold. Since the total sales revenue is $73,785, we can set up the following equation:
40x + 50y + 59[(y + x) + 435] = 73,785
Now, let's simplify the equation:
40x + 50y + 59y + 59x + 25665 = 73,785
Combine like terms:
99x + 109y + 25665 = 73,785
Subtract 25665 from both sides of the equation:
99x + 109y = 48120
We now have a system of equations. The first equation is:
99x + 109y = 48120
The second equation is:
y = 3x + 78
To solve this system of equations, we can use the substitution method or the elimination method. Let's solve it using the substitution method:
From equation 2, we have:
y = 3x + 78
We can substitute this into equation 1:
99x + 109(3x + 78) = 48120
Simplify:
99x + 327x + 8482 = 48120
Combine like terms:
426x + 8482 = 48120
Subtract 8482 from both sides of the equation:
426x = 39638
Divide both sides of the equation by 426:
x = 93
Now, substitute the value of x back into equation 2:
y = 3(93) + 78
Simplify:
y = 279 + 78
y = 357
So, the solution to the system of equations is x = 93 and y = 357.
Therefore, 93 mezzanine tickets, 357 balcony tickets, and (93 + 357 + 435) = 885 main floor tickets were sold.