Theater Revenues A Broadway theater has 500 seats, divided...

Theater Revenues A Broadway theater has 500 seats, divided into orchestra, main, and balcony seating. Orchestra seats sell for $\$ 150,$ main seats for $\$ 135,$ and balcony seats for $\$ 110 .$ If all the seats are sold, the gross revenue to the theater is $\$ 64,250$. If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $\$ 56,750 .$ How many of each kind of seat are there?

Key Concepts

System of Linear Equations

This concept involves setting up multiple equations that represent relationships between unknown quantities. In many real-world applications, like revenue calculations for a theater, these equations are derived from known totals and pricing, and solving the system allows one to determine the specific values of each unknown variable.

Translating Word Problems into Algebraic Equations

This key concept involves reading a real-life scenario and expressing the given information as algebraic equations. It requires identifying variables, understanding constraints such as total counts or overall revenue, and then formulating equations that model the situation.

Elimination and Substitution Methods

These are techniques used to solve systems of equations derived from word problems. The elimination method involves subtracting or adding equations to remove one variable at a time, while the substitution method involves solving one equation for a variable and substituting that expression into another equation, thereby simplifying the system and finding the unknowns.

Constraint Satisfaction

This concept pertains to recognizing and using inherent limitations provided in a problem, such as a fixed total number of items or fixed totals in revenue. Acknowledging these constraints helps in correctly setting up the problem and ensuring that the solutions are valid within the context of the problem.

Transcript

00:01 We're given that a theater has 500 seats.

00:03 There's orchestra seats that sell for 150, main seats itself are 135, and balcony seats itself for 110.

00:11 If all the seats are sold, the revenue would be $64 ,250.

00:17 If all main and balconies are sold but only half of the orchestra, the revenue would be 56 ,750.

00:25 So we do our equations from the first sentence, 500 seats, we got the orchestra seats plus the main seats plus the balcony seats is equal to 500 then we've got our price orchestra is 150 main is 135 balcony is 110 and then all sold gives us this amount of money so 150 times o plus 135m plus 110 b is equal to 64 ,250 and then our last sentence comes of the main and balcony are sold but only half of the orchestra we've got this much amount so 150 times half the orchestras plus 135 ms plus 110 b's equals to 56 seven 150.

01:42 So we simplify.

01:44 So we're going to use our first equation and our second equation.

01:48 We're going to multiply our first equation by, let's see, negative 110.

01:57 And we're going to cancel out our bs.

01:59 So we get negative 110 .0 minus an 110m minus 110b is equal to 110 times 5 .5 .000.

02:14 That equals negative 55 ,000.

02:22 Well, second equation stays the same, 150o plus 135m plus 110b is equal to 64 ,000 250.

02:37 So our bs cancel out.

02:40 We have a little 110 subtracted from 150, gives us 40o.

02:49 And then we have 135 minus 110 because this 25 ms is equal to 65 ,250 minus the 55 ,000, leaves us with 9 ,250.

03:13 So now we do the same thing with the first and the last equation.

03:18 So our first and last equation, draw a line here.

03:22 So we have negative 110 minus, so i forgot the o, minus 110m minus 110b is equal to negative 55 ,000.

03:46 And then our third equation, half of 150 is 75 ,0 plus 135m plus 110b is equal to 56 ,7%.

04:03 So our bs cancel out and we're left with 110 subtracted.

04:16 Left with negative 350 plus 25m is equal to 5675 ,000...

Please give Ace some feedback