Problem 51 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression. $$-8 a+(-15 a)$$

Step-by-Step Solution

Answer

-23a

Step 1: Identify Like Terms

In the expression \(-8 a + (-15 a)\), the terms \(-8a\) and \(-15a\) are like terms because they both have the same variable part, \(a\).

Step 2: Combine Like Terms

Like terms can be combined by adding or subtracting their coefficients. Here, you add -8 and -15 to get -23. This means that -8a + -15a simplifies to -23a.

Key Concepts

Understanding Like Terms

Like terms are terms in an algebraic expression that have the exact same variable parts. This means that both the variable and its exponent must be identical. For example, in the expression \(-8a + (-15a)\), the terms \(-8a\) and \(-15a\) are like terms because they both include the variable \(a\) raised to the same power, which is 1 here. This implies that you can work with them together when simplifying the expression.
In contrast, terms like \(5x\) and \(5x^2\) are not like terms because the exponents of \(x\) are different. Understanding which terms are alike is crucial because it allows you to perform operations such as addition or subtraction on them to simplify an expression.

Introduction to Coefficients

Coefficients are the numerical parts of the terms in an algebraic expression. They tell you how many of each term you have. In the expression \(-8a + (-15a)\), the coefficients are \(-8\) and \(-15\). These numbers indicate how many of the variable \(a\) are being considered.
Whenever you're simplifying expressions, identifying the coefficients is essential. They give you the ability to combine like terms, which is a vital skill in algebra. By operating on the coefficients, you determine the new value of that part of the expression. This mostly involves arithmetic operations like addition, subtraction, multiplication, or division.
Remember: \(-8a\) means \(-8\) times \(a\), so you treat \(-8\) as a separate entity from the variable when performing algebraic operations.

Combining Like Terms for Simplification

Combining like terms is the process of simplifying expressions by adding or subtracting the coefficients of like terms. This is a fundamental step in reducing algebraic expressions to their simplest form.
In the example \(-8a + (-15a)\), since \(-8a\) and \(-15a\) are like terms, you combine them by adding their coefficients: \(-8 + (-15) = -23\). Therefore, the expression simplifies to \(-23a\).
When combining like terms:

• Add or subtract only the coefficients, leaving the variable part unchanged.
• Double-check the signs; negative and positive coefficients can affect your result.

Combining like terms ensures your expression is as simple and compact as possible, which is crucial for solving equations, graphing functions, and more advanced mathematical tasks.