Problem 84 - Introductory Algebra for College Students

Question

Simplify each algebraic expression. $$-6 x+x$$

Step-by-Step Solution

Answer

\(-5x\)

Step 1: Identify the like terms

In this given expression \(-6x + x\), \(-6x\) and \(x\) are like terms because they contain the same variable \(x\).

Step 2: Simplify the like terms

Now, perform the operation. The coefficient of \(x\) in the second term is actually 1 (any term without a coefficient is understood to have 1 as its coefficient), so simply add -6 (coefficient of first term) and 1 (coefficient of second term) to get -5. Once you add those, you obtain: \(-5x\)

Key Concepts

Like Terms

In algebra, recognizing "like terms" is essential for simplifying expressions. Like terms are terms in an expression that have identical variable parts. For instance, in the expression \(-6x + x\), both terms have the variable \(x\), making them like terms. It's crucial to note that the coefficients, or numerical parts of the terms, can differ. Differentiating between like and unlike terms allows us to combine them through addition or subtraction.

• Like terms share the same variables raised to the same powers.
• Pay attention to the whole variable part, including powers.

This simple identification helps in combining terms effectively, a key step in simplifying expressions.

Coefficients

Coefficients are the numerical factors in terms of an algebraic expression. In \(-6x + x\), the numbers -6 and 1 are coefficients. The coefficient shows how many times to multiply the variable. Understanding coefficients is vital because they guide the combining of like terms. Note that:

• If a term lacks an explicit coefficient, it is usually 1.
• Negative signs in coefficients indicate subtraction.

For example, \(x\) has an invisible coefficient of 1, understood during the simplification process.

Simplification

Simplification is the process of reducing expressions into simpler forms, making them easier to work with. After identifying like terms and coefficients, perform the indicated operations to combine them into one term. In the expression \(-6x + x\):

• Add the coefficients of the like terms: \(-6 + 1 = -5\).
• Combine them to form: \(-5x\), a simplified expression.

Through simplification, you make the expression easier to understand and use in further calculations. This streamlined expression maintains identical value and utility.

Variables

In algebraic expressions, variables act as placeholders that can represent numbers or quantities. Here, the variable is \(x\), and it remains consistent throughout the expression \(-6x + x\). Variables allow expressions to be flexible and adaptable, providing a general form that can be adjusted with real numbers. It's important to understand their role:

• Variables remain unchanged during simplification.
• They unify like terms but don't affect coefficient calculations.

Grasping the function of variables helps in deciphering expressions and performing algebraic operations effectively.