Problem 14 - Elementary Algebra Within Reach

Question

In Exercises 11-18, identify the coefficient of the term. $$ \frac{2}{3} n $$

Step-by-Step Solution

Answer

The coefficient of the term \(\frac{2}{3} n\) is \(\frac{2}{3}\).

Step 1: Identify the variable

In \(\frac{2}{3} n\), the letter \(n\) represents the variable.

Step 2: Identify the coefficient

In the term \(\frac{2}{3} n\), the coefficient is \(\frac{2}{3}\), since it is the numerical part that multiplies our variable \(n\).

Key Concepts

Algebraic Expressions

Algebraic expressions are like sentences in mathematics that consist of various components, including numbers, variables, and operators. These expressions can range from simple to complex. For example, in the term \(\frac{2}{3} n\), this is an algebraic expression where \(\frac{2}{3}\) and \(n\) are combined through multiplication.Algebraic expressions are vital because they allow us to express relationships and solve problems efficiently. They can represent real-world situations and are used extensively in fields such as physics, engineering, and economics. Understanding algebraic expressions involves:

• Recognizing each term within the expression.
• Identifying the components like variables and coefficients.
• Understanding the operations connecting them, such as addition, subtraction, multiplication, and division.

Variables in Algebra

Variables are symbols, often letters, used to represent unknown values in algebraic expressions. They are crucial because they allow flexibility and generalization in mathematical problem-solving.In the expression \(\frac{2}{3} n\), the letter \(n\) acts as the variable. It stands in place of a number that we may not know immediately, but can solve for later in an equation.Here are some key points about variables:

• Variables can be any letter from the alphabet. Common examples are \(x\), \(y\), and \(z\).
• In equations or expressions, variables can change or "vary" to assume many values.
• They help in forming general formulas that can solve for unknowns efficiently.

Understanding variables is foundational for solving equations and performing operations like algebraic manipulations.

Numerical Coefficients

A numerical coefficient is the number that appears in front of a variable in an algebraic expression, multiplying it. It is crucial in defining the value of that term.In \(\frac{2}{3} n\), the numerical coefficient is \(\frac{2}{3}\). It tells us how much the variable \(n\) is being multiplied by.Key aspects of numerical coefficients include:

• They indicate the strength or multiplicative factor of a variable in a term.
• Coefficients can be positive, negative, fractions, or whole numbers.
• In equations, knowing the coefficient is essential for solving for the variable efficiently.

By understanding numerical coefficients, we gain better insight into the terms of algebraic expressions and their relative scales and impacts within the expression.