Problem 11 - Prealgebra

Question

Use the distributive property to combine each of the following pairs of similar terms. $$\frac{1}{3}(3 x+6)$$

Step-by-Step Solution

Answer

The expression simplifies to \( x + 2 \).

Step 1: Identify the Expression

The expression given is \( \frac{1}{3}(3x + 6) \). This expression contains the term \( \frac{1}{3} \) outside the parentheses, and inside the parentheses, we have \( 3x + 6 \).

Step 2: Apply the Distributive Property

The distributive property allows us to multiply a term outside the parentheses by each term inside the parentheses. So, we will distribute \( \frac{1}{3} \) to both \( 3x \) and \( 6 \). This gives us two separate multiplications to perform: \( \frac{1}{3} \times 3x \) and \( \frac{1}{3} \times 6 \).

Step 3: Perform the Multiplications

Let's calculate each multiplication separately:- For \( \frac{1}{3} \times 3x \), we multiply the coefficients: \( \frac{1}{3} imes 3 = 1 \), then multiply by \( x \), resulting in \( 1x \) or simply \( x \).- For \( \frac{1}{3} \times 6 \), multiply the coefficients: \( \frac{1}{3} imes 6 = 2 \). This results in the number \( 2 \).

Step 4: Combine the Terms

After performing the multiplications, we combine the results to form the simplified expression. The terms are \( x \) and \( 2 \), so when we add them together, the simplified expression becomes \( x + 2 \).

Key Concepts

Simplifying Expressions

Simplifying expressions is like tidying up a room full of items, making sure everything is neat and organized. It involves breaking down complex mathematical statements into their simplest form, making them easier to work with. In our exercise, we aim to simplify the expression \( \frac{1}{3}(3x + 6) \).To simplify an expression like this, we need to use the distributive property, a fundamental concept in prealgebra. This technique allows us to "distribute" a number (or coefficient) multiplied by a sum within parentheses across each term inside the parentheses. But why do we do this?

• It helps remove parentheses, making calculations and further simplifications easier.
• Simplifying expressions is essential for solving equations and inequalities in higher math concepts.

Let's start by distributing \( \frac{1}{3} \) to each term inside the parentheses, calculating \( \frac{1}{3} \times 3x \) and \( \frac{1}{3} \times 6 \) separately. This breaks down to simpler steps, leading to a clearer expression: \( x + 2 \). After distributing, we further simplify to remove any redundant coefficients or terms, ensuring we have the simplest form.

Combining Like Terms

Combining like terms is a crucial step in simplifying expressions. This process involves merging terms that have the same variable parts. For example, in an expression, all terms with the variable \( x \) are like terms and can be combined.In the exercise given, combining like terms doesn't come into play extensively because the distribution of \( \frac{1}{3} \) results directly in terms that are already as simplified as they can be: \( x \) and \( 2 \). However, understanding this concept is vital for simplifying more complex expressions, where multiple like terms exist.

• Identify terms with the same variables and powers.
• Add or subtract the coefficients of these terms.

Consider an example: If you had \( 2x + 3x \), you could combine them to form \( 5x \). This key step often requires careful attention to detail, as a misstep could lead to errors in solved solutions.

Prealgebra Concepts

Prealgebra is about building a strong foundation for algebra, teaching the skills needed to solve equations and understand algebraic expressions. Concepts such as the distributive property, combining like terms, and simplifying expressions lie at its core.When dealing with problems like \( \frac{1}{3}(3x + 6) \), we rely on these foundational skills.

• Understanding how to manipulate expressions using the distributive property.
• Knowing when and how to combine like terms to simplify expressions further.
• Applying numerical operations and properties correctly.

By honing these prealgebra skills, students prepare themselves for more advanced mathematical problems. They learn to see math as a series of logical steps and rules, which can be broken down into smaller, more manageable parts, providing clarity and consistency in their mathematical journey.