Problem 31 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent algebraic expression. $$3(n+1)$$

Step-by-Step Solution

Answer

The expression \(3(n+1)\) is equivalent to \(3n + 3\).

Step 1: Identify the Expression

The given expression is \(3(n+1)\). This expression has two parts inside the parentheses, \(n\) and \(+1\), which need to be distributed with the number \(3\) outside the parentheses.

Step 2: Apply the Distributive Property

The Distributive Property states that \(a(b+c) = ab + ac\). Here, \(a = 3\), \(b = n\), and \(c = 1\). Apply the property by distributing \(3\) to both \(n\) and \(1\).

Step 3: Multiply First Term

Multiply the first term: \(3 \times n = 3n\). This result represents the distribution of \(3\) over the first term \(n\).

Step 4: Multiply Second Term

Multiply the second term: \(3 \times 1 = 3\). This result represents the distribution of \(3\) over the second term \(1\).

Step 5: Combine the Products

Combine both products from the distribution to form the equivalent expression: \(3n + 3\).

Key Concepts

Understanding Algebraic Expression

An algebraic expression is a combination of numbers, variables, and mathematical operations like addition, subtraction, multiplication, and division. In algebra, an expression serves as a representation of mathematical quantities and relationships.
In the exercise you encountered, the expression is presented as \(3(n+1)\). Here, \(3\) is a number known as a coefficient, \(n\) is a variable, and \(+1\) is a constant term.

• The coefficient (\(3\)) is a number that multiplies the variable(s) in an expression.
• The variable (\(n\)) is a symbol that represents an unknown quantity.
• The constant (\(+1\)) is a fixed value added to the variable.

An essential concept when working with algebraic expressions is understanding how parts of the expression interact with each other, especially through the distributive property.

Exploring Equivalent Expressions

Equivalent expressions are expressions that may look different but represent the same value for any substitution of the variable. The goal of using the distributive property is to convert a given expression into an equivalent form, often simplifying calculations or making the expression easier to work with.
When you distribute \(3\) in the expression \(3(n+1)\), you multiply \(3\) by both \(n\) and \(1\), resulting in \(3n + 3\). This new expression, \(3n + 3\), is an equivalent expression because it will have the same result as the original expression \(3(n+1)\) for any value of \(n\).

• The process of finding equivalent expressions is crucial in algebra to solve equations, simplify expressions, and analyze mathematical scenarios effectively.
• Through equivalent expressions, it's easier to identify and highlight relationships between different quantities in an equation or mathematical model.

Prealgebra Concepts and the Distributive Property

In prealgebra, fundamental concepts are introduced to help students transition from arithmetic operations to more abstract algebraic thinking. One of these key concepts is the distributive property, which aids in simplifying expressions and solving equations.
The distributive property allows you to "distribute" a multiplied value across terms inside parentheses. It's a vital concept because it helps split complex problems into more manageable pieces. For example, in \(3(n+1)\):

• The distributive property helps break down the multiplication into two parts, \(3 \times n\) and \(3 \times 1\).
• This process results in the expression \(3n + 3\), making it clearer and oftentimes easier to work with for further operations.
• While using the property, it also reinforces the skills of multiplication, addition, and understanding of parentheses’ role in governing the sequence of calculations.

The distributive property is not only foundational in prealgebra but a tool that students will repeatedly use in more advanced mathematical contexts.