Theater Revenues A Broadway theater has 500 seats, divided into orchestra, main, and balcony sea

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

Key Concepts

System of Linear Equations

This concept involves setting up two or more equations that relate variables representing unknown quantities. In many word problems, including those concerning revenue or distributions, a system of linear equations is constructed to capture the relationships dictated by the problem's conditions. Solving these systems yields the values of the unknowns.

Variable Assignment and Constraint Formulation

In problems that involve splitting a total into parts or allocating resources, it is essential to assign variables to each unknown category and then express the given conditions as mathematical constraints. These constraints reflect totals, proportional relationships, or other restrictions imposed by the situation.

Revenue Calculation in Word Problems

Revenue calculation in word problems involves relating quantities (like numbers of items sold) with their respective prices to form equations that represent total income. This technique allows one to model and solve problems where overall earnings must be derived from various components, each with its own price.

Equation Solving Techniques

Techniques such as substitution and elimination are applied to solve systems of equations. Once the variables are defined and equations are formulated, these methods are used to systematically reduce the system to find the value of each variable, ensuring consistency with all constraints provided by the problem.

Transcript

00:01 This problem is going to involve the application of solving a system of linear equations.

00:09 In fact, we will have a system of three linear equations and three variables, and we will use the method of elimination to solve the system.

00:20 Now, before we get started with this, let's see a little bit about what this problem involves.

00:25 This problem talks about the seats in a theater and the revenue that the theater brings in from selling so many seats.

00:34 There is a theater that has 500 seats in it.

00:38 We're given the cost of each seat in each of the three sections.

00:42 We're an orchestra section at $50 a seat, a main section at $30 a seat, and a balcony section at $25 a seat.

00:53 Now, if this theater were to sell every single series, it's sold out performance, then the revenue would be $17 ,000.

01:02 $100.

01:04 But if they were to sell only all of the seats in the main section and all the seats in the balcony section and only half of those in the orchestra section, then the revenue is on, it's $14 ,600.

01:21 So we should be able to use the information given to us and figure out how many seats are going to be in each section in this theater.

01:31 So with any application problem, once you determine what you are looking for, you should state your unknowns as variables and what each verbal represents.

01:43 Then use those variables and the information of the problem to write your equations.

01:49 Once you've written the equations, solve the equations, in this case we'll solve the system to answer the problem.

01:58 So what i'm going to start off with is my three unknowns are going to.

02:03 Involve how many seats are in each of the three sections in this theater.

02:08 So i'm going to let x stand for the number of seats in the orchestra section.

02:24 And i will let y be the number of seats in the main section.

02:34 And then z will be the number of seats in the balcony.

02:47 All right, so now i'm going to use the information given to me in the problem and set up equations with these variables.

02:54 All right, so i'll start off and i know.

02:55 There are 500 seats.

02:59 Okay, the problem tells us 500 seats.

03:01 So that means the total number of seats in each of the three sections should add up to 500.

03:08 So my first equation is going to be x plus y plus c since each one of those variables stands for the number of seats in their respective section.

03:20 And i'll set that equal to 500.

03:24 There's my first equation.

03:25 All right, my second equation is going to involve the revenue when we have a full house.

03:33 So if i take what each seat cost in each of the sections times the number of seats in the section, that's the revenue from that particular section.

03:44 Okay, for example, if each seat in the orchestra section costs $50, and if i take 50 times x that represents the revenue if all of the seats in the orchestra section are sold same way 35 y will represent the revenue from the main section and 25 z would be the revenue from the balcony section and again a full house is $17 ,100.

04:20 Then the last bit of information tells us that if half of the orchestra seats are sold but all of the seats in the main and the balcony sections are sold, the revenue is 14 ,600.

04:34 So in a similar way, you say half times the revenue from the orchestra section, all of the revenue from the main section, and all of the revenue from the balcony is $14 ,600.

04:54 Now i'm going to change this first turn to $20 ,000.

04:57 And that will give me my three equations using my three variables.

05:10 For reference purposes as i talk about this problem i'm going to call this equation 1, this one equation 2 and this one is going to be my equation 3.

05:24 Now when you're solving an equation a system of equations with three variables using elimination what you want to do is pick a couple of equations.

05:35 And eliminate one of the variables from them.

05:39 Many times there are many different ways to work such a system.

05:44 It's just kind of what hits you.

05:47 And what gets me when i'm working with this, if i look at equation one and i see the coefficient of x is a positive one, and i look at equation 3 and i see the coefficient of x as a positive 25, i can easily turn this coefficient of a positive 1 into a negative 25 by multiplying the entire equation by negative 25...

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