/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 180 If the number of \(\$ 8\) child ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 180

If the number of \(\$ 8\) child tickets is seventeen less than three times the number of \(\$ 12\) adult tickets and the theater took in \(\$ 584,\) how many of each ticket were sold?

Short Answer

Expert verified
20 adult tickets and 43 child tickets were sold.

Step by step solution

01

Define variables

Let the number of adult tickets be denoted by \( A \) and the number of child tickets be denoted by \( C \).
02

Set up the equations

According to the problem, the relationship between child tickets and adult tickets is given by:\[ C = 3A - 17 \]Additionally, the total revenue from the tickets sold is given by:\[ 12A + 8C = 584 \]
03

Substitute \( C \) into the revenue equation

Replace \( C \) in the total revenue equation with \( 3A - 17 \):\[ 12A + 8(3A - 17) = 584 \]Simplify this equation to:\[ 12A + 24A - 136 = 584 \]
04

Solve for \( A \)

Combine like terms and solve for \( A \):\[ 36A - 136 = 584 \]Add 136 to both sides:\[ 36A = 720 \]Divide by 36:\[ A = 20 \]
05

Calculate \( C \)

Substitute \( A = 20 \) back into the equation \( C = 3A - 17 \):\[ C = 3(20) - 17 \]Simplify to find \( C \):\[ C = 60 - 17 = 43 \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

linear equations
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. In the given exercise, we have two main linear equations. The first equation relates the number of child tickets (C) to the number of adult tickets (A): C = 3A - 17. The second equation considers the total revenue received from selling the tickets: 12A + 8C = 584. Both equations have variables raised to the first power, which makes them linear.
variable substitution
Variable substitution is a method used to solve systems of equations, where you solve one equation for one variable and then substitute that expression into the other equation. In our exercise, we first rearranged the equation C = 3A - 17 to express C in terms of A. Then, we substituted 3A - 17 for C in the second equation: 12A + 8C = 584. By doing this, we transformed the revenue equation into: 12A + 8(3A - 17) = 584, allowing us to solve for A without needing to deal with C directly.
problem solving steps
Breaking down the problem into clear, manageable steps can make it easier to solve. Here are the steps taken in this exercise:
  • Step 1: Define the variables. Let A represent the number of adult tickets and C represent the number of child tickets.
  • Step 2: Set up the equations. Use the information provided to relate A and C and to set up the revenue equation.
  • Step 3: Substitute one variable into the other equation. Solve for one variable in terms of the other to reduce the number of variables.
  • Step 4: Solve for the first variable. Simplify the equation and solve for the remaining variable.
  • Step 5: Solve for the second variable. Use the solution from step 4 to find the value of the other variable.
Following these steps methodically helps streamline the problem-solving process.
algebraic expressions
An algebraic expression is a mathematical phrase that involves numbers, variables, and operations. In our problem, we work with expressions like 3A - 17 and 12A + 8C. Algebraic expressions are used to model real-world situations mathematically. For example, 3A - 17 models the number of child tickets in relation to adult tickets, while 12A + 8C represents the total revenue from ticket sales. By manipulating these expressions, we can solve for unknown quantities and gain insight into the problem.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Christa wants to put a fence around her triangular flowerbed. The sides of the flowerbed are six feet, eight feet and 10 feet. How many feet of fencing will she need to enclose her flowerbed?

In the following exercises, solve. Marcela is registering for her college classes, which cost \(\$ 105\) per unit. How many units can she take to have a maximum cost of \(\$ 1,365 ?\)

Find your last month's phone bill and the hourly salary you are paid at your job. (If you do not have a job, use the hourly salary you would realistically be paid if you had a job.) Calculate the number of hours of work it would take you to earn at least enough money to pay your phone bill by writing an appropriate inequality and then solving it.

In the following exercises, solve. Mona is planning her son's birthday party and has a budget of \(\$ 285\). The Fun Zone charges \(\$ 19\) per child. How many children can she have at the party and stay within her budget?

In the following exercises, solve. Marlon's TV plan costs \(\$ 49.99\) per month plus \(\$ 5.49\) per first-run movie. How many first-run movies can he watch if he wants to keep his monthly bill to be a maximum of \(\$ 100 ?\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.