Problem 30 - Algebra 1

Question

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ -3(r+8) $$

Step-by-Step Solution

Answer

The expression -3(r+8) after using the distributive property and simplifying, becomes -3r - 24.

Step 1: Applying the distributive property

The distributive property states that a(b + c) = ab + ac. Apply this property to the given expression -3(r+8). So the expression becomes -3*r + -3*8.

Step 2: Simplify the expression

Now it is time to simplify the expression. -3*r is -3r and -3*8 is -24. Therefore, the simplified expression is -3r -24

Key Concepts

Algebraic Expressions

Algebraic expressions are like sentences in math, conveying information about numbers and variables. They consist of numbers, variables, and operations combined to express a mathematical relationship. Each part serves a purpose:

• Variables: These are symbols, commonly letters like x, y, or r, used to represent unknown quantities. They allow us to write general expressions that apply in various scenarios.
• Constants: These are fixed numbers within the expressions, such as 5, -7, or 8.
• Operators: These are symbols like +, -, *, or / that tell us which mathematical operations to perform.

An example of an algebraic expression is \(-3(r + 8)\). Here, "r" is the variable, -3 and 8 are constants, and the operation involves multiplication and addition within the parentheses. This combination provides an avenue to express mathematical thoughts clearly and concisely. Understanding how to manipulate these expressions is key to solving many mathematical problems.

Simplification

Simplification is the process of making an algebraic expression easier to understand by removing parentheses and combining like terms. It usually starts with applying certain mathematical properties to reduce complexity. Here’s how this works:1. Remove Parentheses: Use mathematical properties such as the distributive property - which we'll discuss in the next section - to rewrite expressions without parentheses. 2. Combine Like Terms: Terms that have the same variable raised to the same power, such as \(-3r\) and \(-2r\), can be combined. This simplifies the expression further, making it easier to evaluate or solve. For our expression \(-3(r + 8)\), simplification involves distributing the \(-3\) across the addition inside the parentheses. This results in the simpler form \(-3r - 24\). By following these steps, we ensure that expressions are presented in their simplest forms, facilitating easier calculations and insights.

Mathematical Properties

Mathematical properties are rules that help us manipulate numbers and expressions efficiently. Understanding these properties is fundamental to working with algebraic expressions. One of the most important properties in this context is the Distributive Property:- The distributive property allows us to distribute a multiplication operation over addition or subtraction inside parentheses. It is stated as \(a(b + c) = ab + ac\). For instance, in \(-3(r + 8)\), we distribute \(-3\) to both \(r\) and \(8\), resulting in \(-3 * r + -3 * 8\), which simplifies to \(-3r - 24\). This property is incredibly useful for simplifying expressions and solving equations. Other crucial mathematical properties include the Commutative and Associative properties for addition and multiplication, which allow us to change the order or grouping of numbers without affecting the outcome. Familiarity with these principles enhances problem-solving strategies and algebraic manipulation skills.