Problem 44 - Prealgebra

Question

Use the Distributive Property to rewriteh expression as an equivalent algebraic expression. \(\frac{1}{2}(n-9)\)

Step-by-Step Solution

Answer

The equivalent expression is \(\frac{1}{2}n - \frac{9}{2}\).

Step 1: Identify the terms inside the parenthesis

In the expression \(\frac{1}{2}(n-9)\), the terms inside the parenthesis are \(n\) and \(-9\). Our task is to distribute the \(\frac{1}{2}\) to each of these terms.

Step 2: Apply the Distributive Property

The Distributive Property states that \(a(b+c) = ab + ac\). Apply this property to \(\frac{1}{2}(n-9)\). Distribute \(\frac{1}{2}\) to both \(n\) and \(-9\): \(\frac{1}{2} \cdot n + \frac{1}{2} \cdot (-9)\).

Step 3: Simplify Each Term

Calculate each term separately: 1. \(\frac{1}{2} \cdot n = \frac{1}{2}n\)2. \(\frac{1}{2} \cdot (-9) = -\frac{9}{2}\).

Step 4: Write the Equivalent Expression

Combine the simplified terms to write the expression: \(\frac{1}{2}n - \frac{9}{2}\).

Key Concepts

Understanding Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations. It's like a math sentence that represents values and their relationships without using an equals sign. In our exercise, \(\frac{1}{2}(n-9)\) is an algebraic expression because it includes the variable \(n\), the constant numbers \(1/2\) and \(-9\), and the operation within the parenthesis. To interact with these expressions, especially when applying the Distributive Property, it's important to identify all the components. Variables such as \(n\) represent unknown values, while constants are the specific numbers present. Operations like addition and multiplication determine how these components relate to each other.Algebraic expressions are handy because they help in representing and solving real-world problems. They allow flexibility, as you can change the values that variables hold to find different outcomes or solutions to a problem without altering the structure of the expression itself.

Simplifying Expressions Using the Distributive Property

Simplifying an expression means rewriting it in the most reduced form possible while maintaining its original value. The Distributive Property is a crucial tool in this process. It allows you to remove parenthesis by distributing multiplication over addition or subtraction inside them.In the expression \(\frac{1}{2}(n-9)\), using the Distributive Property involves multiplying \(\frac{1}{2}\) by each term inside the parenthesis. The property, stated as \(a(b+c) = ab + ac\), means you perform the multiplication separately for each contained term. Here's how it's done:

• Multiply \(\frac{1}{2}\) with \(n\) to get \(\frac{1}{2}n\).
• Multiply \(\frac{1}{2}\) with \(-9\) to get \(-\frac{9}{2}\).

By doing this multiplication, you simplify the expression to \(\frac{1}{2}n - \frac{9}{2}\), which has no parenthesis. The goal of simplification is to make expressions easier to work with in further calculations.

Creating Equivalent Expressions

Equivalent expressions are different expressions that represent the same value or quantity. Using properties like the Distributive Property, you can transform expressions while preserving their equivalency.For instance, the expressions \(\frac{1}{2}(n-9)\) and \(\frac{1}{2}n - \frac{9}{2}\) are equivalent. Though they look different, applying the Distributive Property reveals that they convey the same mathematical relationship.Understanding equivalency is essential in algebra because it lets you choose the form of an expression that is most convenient for the problem you're solving. Whether you're looking to simplify, solve, or factor expressions, recognizing and creating equivalent forms is a foundational skill in mathematical manipulation.Equivalence ensures flexibility in problem-solving and helps maintain the correctness of mathematical operations across different contexts.