Problem 1 - College Algebra and Calculus: An Applied Approach

Question

Identify the terms of the algebraic expression.\(7 x+4\)

Step-by-Step Solution

Answer

The terms of the algebraic expression \(7 x + 4\) are \(7x\) and \(4\).

Step 1: Identify the operation

An algebraic expression consists of terms separated by operations. The operation in the expression \(7 x + 4\) is addition. Therefore, the elements separated by the addition operation are the terms.

Step 2: Identify the terms

The terms of the expression \(7 x + 4\) are the elements separated by the operation identified in step 1, which is addition. So, the terms in the expression are \(7x\) and \(4\). The term \(7x\) is a product of a constant, 7, and a variable, x. The term \(4\) is a constant.

Key Concepts

Terms in Algebra

When dealing with algebraic expressions, identifying terms is essential. A term is a single component of an expression, which could be a number, a variable, or the product of numbers and variables. In the expression 7x + 4, we have two distinct terms: 7x and 4.

The term 7x is called a variable term because it contains the variable x, meaning its value can change depending on what x represents. The numerical part of this term, known as a coefficient, is 7 in this case. On the other hand, 4 is a constant term because it always represents the same value and doesn't include any variables. The collection of terms in an algebraic expression provides the foundation to perform arithmetic operations and understand the expression's behavior.

Addition Operation

In algebra, the addition operation is one of the fundamental arithmetic operations and is signified by the plus sign (+). It combines numbers, variables, or terms to give a sum total. For instance, if we look at the expression 7x + 4, the addition operation is what brings together the terms 7x and 4.
Understanding addition is crucial when simplifying expressions because it guides us on how to combine like terms. Like terms are terms that have the same variable raised to the same power, which in this case, doesn't apply since 7x is the only term with a variable. However, if we had another term with an x, those could be added together due to the addition operation.

Constants and Variables

Algebraic expressions feature two main components: constants and variables. Constants are fixed numbers that do not change their value. They stand alone, like monuments steadfast in their value within an expression. In the expression 7x + 4, the number 4 is a constant.

Variables, on the other hand, are symbols like x that represent a number, but they are not fixed; they can vary, hence the term 'variable'. They're like chameleons that adjust their 'color' based on what number they're set to represent. In our example, x is a variable within the term 7x. It's essential to differentiate these components as they determine how we interact with the expression during operations like addition, subtraction, and when solving equations.