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Chapter 4: Problem 89

Write a system of two equations in two unknowns for each problem. Solve each system by substitution. Annual play. There were twice as many tickets sold to nonstudents than to students for the annual play. Student tickets were \(\$ 6\) and nonstudent tickets were \(\$ 11 .\) If the total revenue for the play was \(\$ 1540,\) then how many tickets of each type were sold?

Short Answer

Expert verified
55 student tickets and 110 nonstudent tickets were sold.

Step by step solution

01

Define Variables

Let the number of student tickets be denoted by \(s\) and the number of nonstudent tickets by \(n\).
02

Set Up Equations from Problem Statements

From the problem, we know:1. There were twice as many tickets sold to nonstudents as to students: \(n = 2s\).2. The total revenue from the play was \$1540:\t\( 6s + 11n = 1540 \).
03

Substitute the First Equation into the Second

Using \(n = 2s\), substitute \(n\) in the second equation:\t\(6s + 11(2s) = 1540\).
04

Solve for s

Simplify and solve the resulting equation:\t\(6s + 22s = 1540\)\t\(28s = 1540\)\t\(s = \frac{1540}{28}\)\t\(s = 55\).
05

Find n Using the Value of s

Substitute \(s = 55\) back into the first equation \(n = 2s\):\t\(n = 2(55)\)\t\(n = 110\).
06

Verify the Solution

Check the values in the original revenue equation:\t\(6(55) + 11(110) = 330 + 1210 = 1540\).The values satisfy the total revenue.

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Key Concepts

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

Substitution Method
The substitution method is a way to solve systems of equations. It involves replacing one variable with an expression derived from another equation. This helps to simplify the system, making it easier to solve for the unknowns.
Here鈥檚 how it works in our example:
  • We start with two equations:
    1. The relationship between nonstudent and student tickets: \(n = 2s\)
    2. The total revenue equation: \(6s + 11n = 1540\).
  • We substitute the first equation into the second by replacing \(n\) with \(2s\):
    \(6s + 11(2s) = 1540\).
  • This turns our equation into one with only one variable, \(s\), which simplifies our work.
This method is particularly useful when one variable is already isolated or can be easily isolated in one of the equations. It's a systematic way to reduce complexity and find a solution step by step.
Two Unknowns
In the given problem, we have two unknowns: the number of student tickets \(s\) and the number of nonstudent tickets \(n\). We need to find the values of these unknowns that satisfy both conditions set by the problem.
Here鈥檚 how to handle a problem with two unknowns:
  • First, define the variables. In our example, \(s\) represents student tickets and \(n\) represents nonstudent tickets.
  • Next, write down the equations based on the problem's description. For our problem, those equations are:
    1. Nonstudent tickets are twice the student tickets: \(n = 2s\)
    2. Total revenue from both types of tickets: \(6s + 11n = 1540\).
  • Having two equations allows us to solve for both unknowns, making sure the conditions from the problem are met.
Working with two unknowns requires careful management and systematic solving, typically involving methods like substitution or elimination to find the correct pair of values.
Revenue Calculation
Revenue calculation involves finding the total income generated from selling items at specific prices. In our problem, we're calculating the total revenue from selling student and nonstudent tickets.
Here's how we calculate revenue in this context:
  • Each student ticket is priced at \(\text{\textdollar} 6\) and each nonstudent ticket is priced at \(\text{\textdollar} 11\).
  • Our goal is to find how many of each type of ticket were sold to reach a total revenue of \(\text{\textdollar} 1540\).
  • With predefined variables and their relationship, we write the revenue equation: \(\text{\text{\textdollar} 6s + 11n = 1540}\).
  • Substituting the relationship \((n = 2s)\), we get: \(6s + 11(2s) = 1540\).
This allows us to solve for one variable at a time, ensuring our total revenue matches the given amount.
Revenue calculations are common in different scenarios, especially in business contexts, to understand the financial outcomes and ensure targets are met.

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