Theater Revenues A movie theater charges 8.00 for adults, 4.50 for children, and 6.00 for sen

Theater Revenues A movie theater charges 8.00 for adults, ...

Key Concepts

Substitution and Elimination Methods

These are techniques used to solve systems of equations. The substitution method involves solving one equation for one variable and then replacing this expression into another equation, while the elimination method involves adding or subtracting equations to remove a variable, both aiding in reducing the system to a solvable form.

System of Linear Equations

This concept involves forming and solving a set of equations where the unknown variables appear only in the first degree. In problems like these, relationships such as total number of items or total revenue are expressed as linear equations, which can then be solved simultaneously to determine the values of each unknown.

Word Problem Modeling

This concept focuses on translating real-world scenarios into mathematical expressions. It involves identifying the unknown quantities, assigning variables to them, and setting up equations based on the given relationships and constraints, like total counts and sums.

Ratio and Proportion

This concept is used when a problem involves relationships where one quantity is a multiple or a fraction of another. Understanding ratios helps to set up equations correctly, reflecting situations such as one group being twice as large as another.

Transcript

00:01 The objective of this problem is to apply a system of linear equations using the information in the problem to answer what the problem is asking us to find.

00:14 Now this particular problem is going to involve a system of three linear equations with three unknowns.

00:22 Now as with any application problem, you should always determine what your unknowns are, represent them with variables, and then use the information in the problem along with your variables to write your equations.

00:39 We will write a system of three equations with three variables.

00:45 We'll solve that system.

00:47 I'm going to use the method of elimination to solve the system.

00:51 And then we will be able to answer the question asked for in this problem.

00:56 Okay, so let's take a little bit of look at the problem.

01:00 The question in the problem wants to know how many of each type of ticket was sold.

01:07 If we look at the information given about the tickets, these are tickets sold as a movie theater, and it tells us 405 tickets were sold.

01:18 It tells us the revenue that was brought in.

01:21 And it tells us the price of each type of ticket.

01:24 Can i notice there are three types of tickets.

01:27 And then we're also given a little bit of information that relates a couple of the types.

01:33 Now this information is going to allow us to write a system of equations.

01:39 But first i need to determine what my variables are and state what each one represents.

01:46 So what am i doing this problem is i'm going to let x stand for the number of adult tickets sold.

01:54 So i'm just going to say adult tickets sold.

02:07 And i'll let y be the number of children tickets sold.

02:19 Whoops, no, i was going to.

02:22 And z will be the number of senior citizen tickets sold.

02:36 Okay, after i have stated what each verbal represents, then i should be able to use the information in the problem and write my equation.

02:45 So again, the information were told that there were 405 tickets sold.

02:53 So from my very first equation, i'm going to say x plus y plus c is 405.

03:05 Okay, that's the number of adult tickets plus the number of children tickets plus the number of senior citizen tickets is equal to the total of 405.

03:15 Okay, for the revenue, i know that each adult ticket was $8, his child ticket was $4 .50, and each senior ticket was $6, and they had a revenue of $23 .20.

03:32 So what i'm going to do is i'm going to take the number of tickets times the price for that ticket.

03:39 So for the adults at $8 a ticket, x tickets, that accounts for the money that was taken in from selling the adult tickets.

03:53 Okay, in a similar way, i'll take $4 .50 times y, and that's money taken in from the children's tickets.

04:03 And six times z is the money taken in from the senior tickets.

04:08 And the total revenue was 2320.

04:13 So we've got a second equation.

04:17 Now the third equation is going to involve this idea that there were twice as many children tickets as adult tickets sold.

04:27 So i'm going to have the equation.

04:29 The number of children tickets is twice the adult tickets...

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