Problem 30 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent algebraic expression. $$5(y+6)$$

Step-by-Step Solution

Answer

The equivalent expression is \(5y + 30\).

Step 1: Understand the Distributive Property

The Distributive Property states that for any numbers \(a\), \(b\), and \(c\), the expression \(a(b+c)\) is equivalent to \(ab + ac\). This means we need to multiply each term inside the parenthesis by the term outside.

Step 2: Apply the Distributive Property

In the expression \(5(y+6)\), we will apply the Distributive Property. This means multiplying 5 by \(y\) and 5 by 6, resulting in the terms \(5y\) and \(5 \times 6\).

Step 3: Perform the Multiplication

Calculate each of the multiplication operations: \(5 \times y = 5y\) and \(5 \times 6 = 30\).

Step 4: Combine the Terms

Combine the results from the multiplications into a single expression: \(5y + 30\). This is the simplified expression after applying the Distributive Property.

Key Concepts

Equivalent Expressions

In algebra, equivalent expressions are different expressions that represent the same value, regardless of the variables' value chosen. This is a fundamental concept because it helps us verify that our transformations and manipulations of expressions maintain the integrity of the original mathematical statement.

Let's consider the expression provided in the exercise: \(5(y + 6)\). By utilizing the Distributive Property, it was transformed into \(5y + 30\). These two expressions are said to be equivalent because, for any value of \(y\), both expressions will yield the same result.

A key point about equivalent expressions is:

• They help to simplify complex expressions.
• They allow us to rewrite expressions in a more manageable or more understandable form.

Recognizing and creating equivalent expressions is a cornerstone of algebra that is widely applied, especially when solving equations and simplifying expressions.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and arithmetic operations. In the exercise example, \(5(y + 6)\), we see a mix of a number (5), a variable \(y\), and an arithmetic operation (addition).

Algebraic expressions are fundamental in mathematics because they form the basis for equations and inequalities. Here are a few characteristics to remember:

• They don't have an equals sign. Once you have an equals sign, you're working with an equation.
• They can be simplified or expanded using properties such as the Distributive Property.

Comprehending algebraic expressions involves understanding how different components interact within the expression and recognizing how properties like the distributive property allow us to manipulate these expressions.

Simplifying Expressions

Simplifying expressions means transforming them into their simplest or most efficient form without changing their value. In the context of the exercise, simplifying involved using the Distributive Property to transform \(5(y + 6)\) into \(5y + 30\). This simplification process often involves operations such as combining like terms or removing parentheses by distributing factors.

The purpose of simplifying expressions is to make them easier to work with or understand. Here are a few reasons why simplification is useful:

• It makes analyzing and solving equations or problems easier.
• It helps identify equivalent expressions quickly.
• Simplifying can reveal properties of the function or expression that might not be immediately apparent.

Simplifying expressions is a skill that sharpens your ability to manipulate and understand mathematical phrases accurately. It is foundational in algebra and crucial for solving more complex problems efficiently.