Problem 49 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The equivalent expression is \(3a + 3b\).

Step 1: Identify the Original Expression

The original expression given is \(3(a+b)\). This expression involves the application of the Distributive Property in algebra.

Step 2: Apply the Distributive Property

The Distributive Property states that for any numbers \(a\), \(b\), and \(c\), \(a(b+c) = ab + ac\). Here, we need to distribute the '3' to both 'a' and 'b' inside the parentheses.

Step 3: Distribute 3 to Each Term

Multiply 3 by 'a' to get \(3a\) and multiply 3 by 'b' to get \(3b\). This step involves performing the multiplication as follows: \(3 \times a\) and \(3 \times b\).

Step 4: Write the Equivalent Expression

After distributing and multiplying, the expression becomes \(3a + 3b\). This is the equivalent algebraic expression using the Distributive Property.

Key Concepts

Equivalent Algebraic Expression

In algebra, an equivalent algebraic expression means rewriting an expression in a different form without changing its value. Imagine you have two coins, both worth 25 cents. Whether you call it two quarters or 50 cents, the value remains the same. Similarly, in our exercise of using the Distributive Property, the expression \(3(a+b)\) transforms into \(3a + 3b\) while retaining its original value. This technique allows us to manipulate expressions for easier computation or simplification, yielding expressions that are 'equivalent,' meaning they hold identical mathematical value or meaning, but look different.

Multiplication

Multiplication is a fundamental operation in mathematics, often described as repeated addition. It's a key part of the Distributive Property and algebraic expressions. In our original problem, we use multiplication to apply the distributive property.

• Multiply 3 and \(a\) to get \(3a\).
• Multiply 3 and \(b\) to get \(3b\).

This demonstrates how multiplication is not just a straightforward operation, but a tool for transforming expressions, which is essential in simplifying and rewriting them, making calculations more efficient and manageable.

Algebra

Algebra introduces letters or symbols, known as variables, to represent numbers. This allows for generalizations and the solving of equations. It's like solving a puzzle where letters stand for unknown pieces.The expression \(3(a+b)\) utilizes the variable letters 'a' and 'b' which can represent any numbers. By practicing algebra, students learn not just to solve for these unknowns, but also to manipulate expressions into forms that are more convenient to work with.Algebra is versatile, allowing you to understand relationships between quantities and solve for unknown variables. This foundational concept is fundamental in advancing to more complex mathematical topics.

Step by Step Solution

Providing a step by step solution helps demystify the process of solving an algebraic expression, ensuring clarity at each phase.

• **Step 1: Identify the Expression** - Recognize what you are working with. In our case, \(3(a+b)\).
• **Step 2: Apply the Distributive Property** - Know the rule: \(a(b+c) = ab + ac\).
• **Step 3: Distribute and Multiply** - Assign '3' to both 'a' and 'b', calculating \(3a + 3b\).
• **Step 4: Rewrite the Expression** - Combine your results into the new equivalent expression: \(3a + 3b\).

This breakdown offers transparency at each stage, enhancing understanding and making such transformations intuitive.