Problem 41 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The equivalent expression is \(6r - 30\).

Step 1: Understand the Distributive Property

The Distributive Property states that multiplying a number by a sum or difference is the same as multiplying the number by each addend or minuend/subtrahend and then adding or subtracting the products. It can be expressed as: \( a(b + c) = ab + ac \) or \( a(b - c) = ab - ac \).

Step 2: Apply the Distributive Property

Apply the Distributive Property to the given expression \((r - 5)6\): multiply \(6\) by each term inside the parentheses. This gives: \[(r - 5)6 = 6 \cdot r - 6 \cdot 5\]

Step 3: Simplify the Expression

Perform the multiplication for each term: \(6 \cdot r = 6r\) \(6 \cdot 5 = 30\) Combine these results to get the equivalent algebraic expression: \[6r - 30\]

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operations. They are essential in prealgebra and advanced math, as they form the language that represents real-world problems in mathematical terms.
A typical algebraic expression might look like it has simple numbers, variables like \( x \), \( y \), or even symbols like \( r \) in your exercise. These components are linked together using mathematical operations such as addition, subtraction, multiplication, and sometimes division.
When working with algebraic expressions, one of the key skills is to manipulate them to simplify or reformat the expression, without changing its value. This is where properties like the Distributive Property come in handy.

Equivalent Expressions

Equivalent expressions are different expressions that represent the same quantity. Understanding this concept is crucial for solving algebraic equations and simplifying expressions.
Equivalent expressions essentially mean that even if two expressions may look different at first glance, they hold the same value. For example, in your exercise,

• The expression \((r - 5)6\) can be rewritten as \(6r - 30\).
• Even though they look different, both expressions will yield the same result for any value assigned to \( r \).

Recognizing and creating equivalent expressions allow you to solve equations more easily. You can interchange expressions freely as long as they are equivalent, making calculations and problem-solving more efficient.

Prealgebra

Prealgebra sets the foundation for all future algebra studies and includes the fundamental concepts necessary for understanding algebraic expressions and properties.

• It introduces students to basic arithmetic involving variables, helping them transition from arithmetic operations with specific numbers to more generalized operations involving unknowns.
• Key topics covered in prealgebra include understanding variables, learning properties like the Distributive Property, and grasping the concept of equivalent expressions.

The goal of prealgebra is to build a strong number sense, basic problem-solving strategies, and the ability to work comfortably with expressions and variables. This foundational knowledge is crucial for tackling more complex problems as you advance in mathematical studies.