Problem 81 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(77-96,\) simplify each algebraic expression. $$8 x+x$$

Step-by-Step Solution

Answer

The simplified form of the expression \(8x + x\) is \(9x\).

Step 1: Identify like terms

In the expression \(8x + x\), we can see that both of the terms are 'like terms' which means they all include the same variable, x.

Step 2: Combine like terms

The like terms can be combined by adding their coefficients. When there is no numerical coefficient with a variable, it is understood to be 1. So we can write the expression as \(8x + 1x\).

Step 3: Perform the addition

Now, add the coefficients of the like terms to get: \(9x\).

Key Concepts

Combining Like Terms

Understanding how to combine like terms is crucial for simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. For instance, in the expression given in our exercise, both terms contain the variable 'x' and they are to the first power, meaning they are like terms.

Simplifying an expression by combining like terms involves adding or subtracting their coefficients. A coefficient is the numerical part of a term that is multiplied by the variable. In the solution process, we started by identifying the like terms in the expression, which were '8x' and 'x', and recognized that they indeed can be combined.

When combining them, remember that if there's no coefficient written explicitly, it's an implicit 1. Thus, 'x' is actually '1x'. By adding up the coefficients 8 and 1, we easily find that '8x + x' simplifies to '9x'. This principle applies to any combination of like terms, making it a foundational tool for algebraic manipulation.

Algebraic Coefficients

Algebraic coefficients are the multipliers of the variables in algebraic terms. They provide valuable information about the term's relationship to the variable it is associated with. In the given problem, the term '8x' has 8 as its coefficient and the term 'x' implicitly carries a coefficient of 1. Recognizing these coefficients is the first step to successfully combining like terms.

It is also important to note that coefficients can be positive or negative, which influences the operation when combining terms. If the exercise had terms like '-8x + x', the coefficient for 'x' would still be considered as positive 1, leading to a simplified expression of '-7x'.

Understanding the role of algebraic coefficients is vital as it helps in performing arithmetic operations on algebraic expressions. They allow us to predict the behavior of the variable term as a whole when the variable's value is known or changed.

Intermediate Algebra

Intermediate algebra serves as a bridge between basic algebraic concepts and more complex mathematical analysis. It involves understanding and manipulating expressions and equations to find the values of unknown variables or to simplify the expressions themselves. The problem of simplifying the expression '8x + x' falls under this category.

In intermediate algebra, we are often faced with tasks like factorization, working with rational expressions, and dealing with quadratic equations, among others. Combining like terms, as demonstrated in the solution to the given problem, is a fundamental skill that you will use repeatedly in this field. Mastery of this topic allows you to efficiently simplify expressions, which is an important step in solving more intricate algebraic problems.

This discipline requires an understanding of various rules and properties of operations—such as the distributive property, the associative property, and the commutative property—to handle expressions and equations efficiently. By mastering these principles, students can move on to more challenging algebraic concepts with a solid foundation.