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Chapter 6: Problem 18

Neil sells subscriptions by phone. He makes \(\$ 2.10\) for each subscription sold. At the holiday time last year, he sold \(n\) subscriptions. Express his earnings algebraically.

Short Answer

Expert verified
Neil's earnings based on the number of subscriptions he sold can be expressed algebraically as Earnings = \$2.10 * n.

Step by step solution

01

- Identify Variables

The problem provides two significant variables. The first one is the amount Neil makes from selling a single subscription, which is \$2.10, and the second one is the number of subscriptions sold, which is 'n'. Our job is to connect these variables in an equation.
02

- Formulate the Equation

Neil earns \$2.10 for each subscription sold. Therefore, if he sells 'n' subscriptions, his total earnings would be the product of the per subscription amount and the number of subscriptions sold. So his earnings can be expressed as: Earnings = \$2.10 * n.
03

- Simplify the Equation

The equation Earnings = \$2.10 * n is a simple and correct representation of Neil's earnings based on the number of subscriptions he sold. This equation need not be simplified any further as it already expresses the relationship in its simplest form.

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Key Concepts

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

Understanding Variables in Algebra
In algebra, a **variable** is a symbol or letter that represents a number. The number can vary or change, which is why it's called a variable. Variables are used to create expressions or equations, making it possible to solve problems with unknown amounts. In our example with Neil, the variable is 'n', which stands for the number of subscriptions sold. By using 'n', we can write equations that accurately express Neil's earnings without knowing exactly how many he sold yet.
  • Variables allow for generalization in mathematical solutions.
  • They can be any letter, such as 'x', 'y', or 'n'.
  • It's essential to define what your variable represents for clarity.
Understanding variables is a fundamental skill in algebra that helps simplify and solve real-world problems.
Linear Equations and Their Structure
A **linear equation** is an equation where the highest power of the variable is one. This means the variable is not squared, cubed, etc. Linear equations form straight lines when graphed, hence the name 'linear.' In Neil's case, the equation that describes his earnings is linear: \[ Earnings = 2.10 imes n \] This tells us his total earnings are **directly proportional** to the number of subscriptions he sells.
  • Linear equations often have one or two variables.
  • They can have a constant term, but in Neil鈥檚 case, it directly depends on 'n'.
  • These equations are foundational for understanding more complex algebraic concepts.
Linear equations are simple yet powerful tools that describe constant rates of change and are widely used in various fields.
Exploring Basic Algebra Concepts
**Basic algebra concepts** form the building blocks of problem-solving in mathematics. Algebra allows us to work with unknown values and develop methods to find these values. Key concepts include:
  • Expressions: Combinations of numbers, variables, and operators (such as addition or multiplication), like the expression \( 2.10 imes n \). Expressions do not have an equal sign.
  • Solving Equations: Finding the value of the variable that makes the equation true.
  • Substitution: Replacing variables in expressions or equations with actual numbers to calculate a specific result.
Basic algebra enables us to create models for real-life scenarios, like calculating earnings from sales. It facilitates understanding of how different quantities relate to each other, demonstrating why algebra is a critical tool in mathematics.

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Most popular questions from this chapter

Ron earns x dollars per hour. He regularly works 40 hours per week. Express his annual salary algebraically.

Mike works at Cheesecake King. He earns \(11 per hour as a busboy. The waiters he helps give him 25% of their tips. a. If Mike worked 6 hours today, how much did he earn, without tips? b. The waiters Mike assisted waited on 16 tables, and the total bill from all these tables was \)1,188. The waiters earned 15% in tips, and gave 25% of these earning to Mike. How much did Mike make in tips? c. What was Mike鈥檚 total salary for the day? d. What were Mike鈥檚 average earnings per hour, including tips? Round to the nearest cent.

Roger wants to have 400 copies of his resume printed. His local print shop charges \(\$ 21.50\) for the first 200 copies and \(\$ 10\) for every 100 additional copies. a. How much will the 400 copies cost, including a sales tax of 6\(\% ?\) b. If the number of sets of 100 resumes is represented by \(x,\) express the cost of the resumes, \(r(x)\) , as a piece wise function of \(x .\)

When Lisa started at her current job, her employer gave her two days of paid vacation time with a promise of three additional paid vacation days for each year she remains with the company to a maximum of four work weeks of paid vacation time. a. Let x represent the number of years she has worked for this employer and y represent the number of paid vacation days she has earned. Write an equation that models the relationship between these two variables. b. It has been five years since Lisa began working for this employer. How many paid vacation days has she earned? c. When will she reach the maximum number of paid vacation days allowed?

Aileen is a salesperson at Lopez Sporting Goods. She is paid a monthly commission on all Little league uniforms she sells. She receives 10\(\%\) of her first \(\$ 2,000\) in sales and 12\(\%\) of the balance of her sales. Last week she earned \(\$ 231.20 .\) What was the total value of the uniforms she sold?
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