a 6,000 seat theater has tickets for sale at $26 and $40 . How many tickets must be sold at each price for a sell out performance to generate a revenue of $190500?
if x sell at $40, then the rest (6000-x) are $26, so
40x + 26(6000-x) = 190500
To determine the number of tickets that must be sold at each price, let's assign variables to represent the number of tickets sold at each price. Let's say x represents the number of tickets sold at $26 and y represents the number of tickets sold at $40.
According to the problem, the total revenue from ticket sales is $190,500. We can express this mathematically as:
26x + 40y = 190500
The next step is to use the information about the total number of seats available in the theater. The theater has a capacity of 6,000 seats, which means the total number of tickets sold should be equal to the number of seats in the theater:
x + y = 6000
We now have a system of equations:
26x + 40y = 190500
x + y = 6000
We can use elimination or substitution method to solve this system of equations.
Since the second equation is already solved for x, we can substitute that value into the first equation:
26x + 40(6000 - x) = 190500
Now, let's solve for x:
26x + 240000 - 40x = 190500
-14x = -49500
x = -49500 / -14
x = 3500
Now that we have the value of x, we can substitute it back into the second equation to find y:
3500 + y = 6000
y = 6000 - 3500
y = 2500
Therefore, to generate a revenue of $190,500, 3500 tickets must be sold at $26 each, and 2500 tickets must be sold at $40 each.