Problem 61 - Algebra 1: Concepts and Skills

Question

SIMPLIFYING EXPRESSIONS Simplify the expression. (Lesson 2.7) $$ 5(y+3)+7 y $$

Step-by-Step Solution

Answer

The simplified form of the expression \(5(y+3) + 7y\) is \(12y + 15\).

Step 1: Distribute

First, apply the distributive property to the expression \(5(y + 3)\). Multiply 5 with each term in the parentheses to result in \(5y + 15\). So, the expression becomes \(5y + 15 + 7y\).

Step 2: Combine Like Terms

Next, combine the like terms from the expression. The like terms are the terms that have the same variable. In this case, the like terms are \(5y\) and \(7y\). Adding these together gives \(12y\). So, the simplified expression now becomes \(12y + 15\).

Key Concepts

Distributive Property

The distributive property allows you to simplify expressions by distributing, or multiplying, a single term across terms within parentheses. This property is fundamental in algebra and understanding it can simplify complex expressions.For example, in the expression \(5(y+3)\), you apply the distributive property by multiplying 5 by each term inside the parentheses:\[5 imes (y) + 5 imes (3) = 5y + 15.\] This step clears the parentheses and turns the expression into a simpler form that is easier to work with.Remember, to apply this property:

• Identify the term outside the parentheses.
• Multiply it with each term inside the parentheses.
• Simplify the resulting expression.

This method not only assists in simplification but also aids in developing a clear understanding of how variables and constants interact in algebraic expressions.

Combining Like Terms

Combining like terms is a powerful technique used to simplify algebraic expressions by merging terms with the same variable.In the expression \(5y + 15 + 7y\), identify like terms: both \(5y\) and \(7y\) are 'like' because they share the same variable, \(y\). These terms can be added together as they represent the same quantities but in different amounts. When you add them, you get \(5y + 7y = 12y\).To effectively combine like terms, follow these steps:

• Look for terms in the expression that have the same variable part.
• Add or subtract the coefficients of these terms.
• Rewrite the expression with combined terms for a final simplified version.

Combining like terms reduces complexity and makes it easier to analyze or further manipulate an expression.

Algebraic Expressions

Algebraic expressions play a significant role in mathematics, representing quantities that can change. They consist of variables, numbers, and operations:

• Variables (e.g., \(y\)) represent unknown numbers.
• Coefficients are the numbers that multiply variables (e.g., the '5' in \(5y\)).
• Constants are standalone numbers without variables (e.g., '15').

Expressions like \(5y + 15 + 7y\) combine these elements to form equations used in various applications.Understanding algebraic expressions involves recognizing parts and how they interact through operations like addition, subtraction, and multiplication.By simplifying expressions, as seen in the original problem, we translate complex equations into manageable forms, essential for problem-solving in algebra and beyond.