Problem 6 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(1-14\) by combining similar terms. $$6 n+13 n-15 n$$

Step-by-Step Solution

Answer

The expression simplifies to \(4n\).

Step 1: Identify Like Terms

In the expression \(6n + 13n - 15n\), all terms contain the variable \(n\). Therefore, they are like terms and can be combined.

Step 2: Combine Like Terms

Add and subtract the coefficients of the like terms: \(6 + 13 - 15\).

Key Concepts

Combining Like Terms

When working with algebraic expressions, the process of combining like terms is essential. It means grouping and simplifying terms that have the same variables and powers. In the expression \(6n + 13n - 15n\), each term includes the variable \(n\). Thus, these are like terms, and their coefficients can be combined.

Here's the step-by-step process of combining these terms:

• Identify and group the like terms, which in this case is anything with the variable \(n\).
• Add or subtract the coefficients (the numbers in front of the variable). For our example, calculate \(6 + 13 - 15\).
• The result simplifies the expression to \(4n\).

Through combining like terms, expressions become simpler and easier to manage. This is particularly useful when solving equations, as it reduces complexity and allows you to focus on solving the problem.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations like addition or subtraction. In the context of our problem, the expression \(6n + 13n - 15n\) combines these elements.

Algebraic expressions do not have an equal sign like equations. They are used to represent relationships or formulas. Key components of expressions include:

• Variables: Letters (such as \(n\)) that stand in for unknown or variable quantities.
• Coefficients: Numbers multiplied by the variables (like 6 in \(6n\)).

Recognizing these parts helps in effectively simplifying and solving algebraic expressions. Understanding how to manipulate them is fundamental to algebra and prepares students for more complex problem-solving.

Coefficients

Coefficients are a crucial component of algebraic expressions. They are the numbers attached to the variable part of the term and essentially represent how many times the variable is being considered.

In the expression \(6n + 13n - 15n\), the numbers 6, 13, and -15 are coefficients. These values tell us how many \(n\)'s we're working with:

• 6 is the coefficient of the first term, indicating 6 copies of \(n\).
• 13 is the coefficient of the second term.
• -15 is the coefficient of the third term, which contributes negatively.

To combine like terms, you only need to focus on adding or subtracting these coefficients, as they determine the weight or emphasis of the term in the expression. Grasping the role of coefficients allows for efficient simplification and problem resolution in algebra.