Problem 51 - Intermediate Algebra

Question

Eric has rock and country songs on his playlist. The number of rock songs is 14 more than twice the number of country songs. Let \(c\) represent the number of country songs. Write an expression for the number of rock songs.

Step-by-Step Solution

Answer

The expression for the number of rock songs is \(2c + 14\).

Step 1 - Understand the given information

Eric has rock and country songs on his playlist. The number of rock songs is 14 more than twice the number of country songs. Let the variable \(c\) represent the number of country songs.

Step 2 - Identify the relationship

The relationship between the rock songs and country songs is given: The number of rock songs is 14 more than twice the number of country songs. This can be written in mathematical terms.

Step 3 - Formulate the expression

Twice the number of country songs is represented as \(2c\). Since the number of rock songs is 14 more than this value, add 14 to \(2c\), giving the expression \(2c + 14\).

Step 4 - Write the final expression

Therefore, the expression for the number of rock songs in terms of the number of country songs \(c\) is \(2c + 14\).

Key Concepts

Variables in Algebra

In algebra, variables are symbols that represent unknown values. These symbols, usually letters, help us create mathematical expressions and equations. In the given exercise, the variable is \(c\) which stands for the number of country songs Eric has on his playlist.

Here are some key points about variables in algebra:

• Variables can represent different quantities.
• They allow us to generalize and solve problems in a systematic way.
• Commonly used letters for variables include \(x\), \(y\), and \(z\) but any letter can be used.

Variables help translate real-world problems into algebraic expressions, making it easier to understand and solve them. By using \(c\) to stand for the number of country songs, we can create an expression to represent the number of rock songs more easily.

Linear Relationships

A linear relationship showcases a straight-line connection between two variables. In this exercise, the relationship between rock and country songs is linear since it follows a direct pattern. Understanding the concept of linear relationships is crucial for analyzing and interpreting data.

Key characteristics of linear relationships include:

• They can be represented by linear equations like \(y = mx + b\).
• They depict a constant rate of change.
• The graph of a linear relationship is always a straight line.

In Eric's playlist, the number of rock songs directly depends on the number of country songs according to the equation \(r = 2c + 14\). This linear expression shows that for each additional country song, the number of rock songs increases predictably according to this formula.

Writing Expressions

Writing expressions is about translating real-world problems into algebraic terms. This skill allows us to represent complex scenarios with mathematical clarity.

Here's how we can write expressions from word problems:

• Identify the quantities involved and choose variables to represent them.
• Understand the relationships between the quantities.
• Translate the relationships into mathematical operations.

In the exercise, we started by letting \(c\) represent the number of country songs. We then identified that the number of rock songs is 14 more than twice the number of country songs. By translating this relationship into an expression, we arrived at \(2c + 14\). This algebraic expression succinctly captures the given information and helps us easily calculate the number of rock songs for any number of country songs.