Theater Revenues A movie theater charges 11.00 dollar for...

Name: Problem 80, Chapter 11: Precalculus
Uploaded: 2021-05-30T23:21:50-08:00
Duration: 4 min 26 s
Channel: Erika Bustos
Description: Theater Revenues A movie theater charges 11.00 dollar for adults, 6.50 dollar for children, and 9.00 dollar for senior citizens. One day the theater sold 405 tickets and collected 3315 dollar in receipts. Twice as many children's tickets were sold as adult tickets. How many adults, children, and senior citizens went to the theater that day?

Theater Revenues A movie theater charges 11.00 dollar for adults, 6.50 dollar for children, and 9.00 dollar for senior citizens. One day the theater sold 405 tickets and collected 3315 dollar in receipts. Twice as many children's tickets were sold as adult tickets. How many adults, children, and senior citizens went to the theater that day?

Key Concepts

System of Linear Equations

This concept involves setting up a collection of equations that represent multiple relationships among unknown quantities. In many real-world problems, such as ticket sales or budgeting, several relationships can be modeled simultaneously, and solving the system provides the values of the unknowns.

Translating Word Problems into Equations

This key concept focuses on reading a problem carefully and translating the information into mathematical expressions. It requires identifying unknown quantities, assigning variables, and constructing equations based on sums, totals, ratios, or other relationships described in the problem.

Incorporating Ratio and Proportional Relationships

Many problems involve ratios that relate different quantities. Recognizing and properly formulating these relationships is crucial. It allows for reducing the number of unknowns and simplifying the system of equations by expressing one variable in terms of another.

Substitution Method

This is a common technique used to solve systems of equations, where one equation is rearranged to express one variable in terms of another and then substituted into the other equations. This method helps in reducing the number of variables and solving the system step-by-step.

Transcript

00:01 We're given a movie theater charged $11 for adult tickets, $6 .50 for children tickets, $9 for senior tickets.

00:07 They sold 405 total tickets for 3 ,315.

00:12 We also know that there's twice as many children tickets were sold as adult tickets, and we need to find how many of each were sold.

00:20 So our first sentence comes for the total number of tickets, and we know we have adults, children, and seniors.

00:27 So adults plus children.

00:30 Plus seniors is equal to 405 tickets.

00:34 Then we have our money.

00:36 We have $11, $6 .50, $9 for a total of 3 ,315.

00:43 So each adult ticket was 11 plus each child's ticket was 650, plus each senior ticket was 9, for a total of 3 ,315.

00:55 And so our last equation comes from twice as many children tickets as adult tickets, so our children's tickets were twice whatever the adults were sold.

01:05 So now we can take that children's ticket and substitute it into the c in the second equation and the c in the first equation.

01:17 So we rewrite those two equations.

01:19 We have adults plus two adults plus seniors is equal to 405.

01:27 So that simplifies to be 3a plus s is equal to 405.

01:33 Our second equation is 11 adults plus 650 times 2a plus 9 seniors is equal to 3315.

01:46 So this becomes 11 adults, 650 twice.

01:51 Comes out to be 13a plus 9s is equal to 3315.

02:00 So we combine like terms.

02:02 So 11 plus 13 comes out to be 24.

02:07 A plus 9s equals 3315.

02:13 So now we use this equation and this equation to solve for one of the variables.

02:20 So we're going to multiply the 3a plus s equation by negative 9.