Problem 11 - Elementary Algebra Within Reach

Question

In Exercises 11-18, identify the coefficient of the term. $$ 14 x $$

Step-by-Step Solution

Answer

The coefficient of the term \( 14x \) is 14.

Step 1: Understanding the Term

The term given is \( 14x \). It's a simple term consisting of a number (the coefficient) multiplied by a variable.

Step 2: Identifying the Coefficient

The number that directly multiplies a variable in a term is called the coefficient. Here, the coefficient is the number '14'.

Key Concepts

Algebraic Terms

An algebraic term is a foundational concept in algebra that represents a piece of an algebraic expression. It is composed of numbers, variables (which are symbols representing unknown values), and sometimes exponents that are all multiplied together. For instance, in the term \( 14x \), \( 14 \) is a constant that multiplies the variable \( x \), making it an algebraic term. Understanding algebraic terms is crucial for manipulating and simplifying expressions, as well as for solving algebraic equations.

An algebraic term can be a single number (a constant), a single variable (like \( x \) or \( y \)), or a combination of both. Terms are separated by plus \( + \) or minus \( - \) signs in an expression. For instance, in \( 3x^2 - 2x + 7 \) there are three terms: \( 3x^2 \), \( -2x \), and \( 7 \). Recognizing and understanding each term's structure is vital for further algebraic calculations.

Variables in Algebra

In algebra, variables are symbols that represent unknown values and are typically denoted by letters such as \( x \) or \( y \). These variables are one of the cores in algebraic expressions, equations, and functions. They can take on various values, and the objective when solving algebraic equations is to find the value of these variables that makes the equation true.

Variables can represent any number—whether it's a whole number, a negative number, a fraction, or even an irrational number. The beauty of variables lies in their ability to generalize mathematical relationships and to allow for the expression of a broad range of equations and formulas that can then be solved in a variety of scenarios, making algebra both versatile and practical in problem-solving.

Multiplication in Algebra

Multiplication in algebra is the process of combining coefficients with variables or other numbers. It follows the same basic rules as multiplication in arithmetic, where numbers are combined to get a product. In algebra, these numbers can include whole numbers, fractions, decimals, and negative numbers, and they can multiply variables or other expressions.

For example, \( 14x \) illustrates basic multiplication where the number \( 14 \) (coefficient) multiplies the variable \( x\). If you encounter a term like \( 7x \) multiplied by \( 2y \)—expressed as \( 7x \cdot 2y \)—you would multiply the coefficients (7 and 2) to get \( 14 \) and then multiply the variables \( x \) and \( y \) to get \( xy \) resulting in \( 14xy \) as the product. This process, known as the distributive property or 'foil' method when dealing with binomials, is fundamental for expanding expressions and solving algebraic equations.