Problem 4 - Algebra 1: Concepts and Skills

Question

Explain how you would use the distributive property to simplify the expression. $$ (x-6) 4 $$

Step-by-Step Solution

Answer

By applying the distributive property, the expression \((x-6) 4\) simplifies to \(4x - 24\)

Step 1: Identifying the Elements

First, identify the elements in the expression that need to be distributed. In the given expression, the number \(4\) is being multiplied by the expression in parentheses \((x - 6)\).

Step 2: Applying the Distributive Property

Next, apply the distributive property, which states that \(a(b + c) = ab + ac\) and similarly \(a(b - c) = ab - ac\). Subsequently, multiply the number outside the parentheses by each term inside the parentheses.

Step 3: Simplified Expression

After distributing, the expression would simplify to \(4x - 24\)

Key Concepts

Simplifying Expressions

Simplifying expressions is a core part of algebra that helps make complex equations more manageable. The goal is to reduce expressions into a simpler form. This involves removing parentheses and combining like terms wherever possible. It makes equations easier to work with and understand. For instance, when we work with a problem like \[(x-6)4\]we aim to break it down to its simplest form. When simplifying through the distributive property, we multiply each term in the parentheses by the number outside. This ensures each element is accounted for appropriately. Often, simplifying will include:

• Distributing numbers across terms in parentheses
• Combining like terms after distributing
• Rewriting expressions cleanly

Simplifying is especially helpful in solving equations, as it allows us to see the underlying relationships between numbers and variables without excessive clutter.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetical operations. They help us represent mathematical ideas in a concise, formulaic manner. In algebra, expressions don't have an equal sign as equations do. Instead, they serve as a statement showcasing a relationship between different elements.Consider the expression \[(x-6)4\]Here,

• 'x' is a variable representing an unknown number, serving as a placeholder
• '6' is a constant which stays the same
• '4' is a coefficient, multiplying the entire expression within the parentheses

Working with algebraic expressions usually involves knowing how to manipulate these components to simplify or solve algebra-related problems. A clear understanding of the components helps in applying rules, like the distributive property, effectively.

Multiplication in Algebra

Multiplication in algebra follows the standard arithmetic rules but often involves variables and coefficients. It's crucial in forming and simplifying algebraic expressions. Understanding how to multiply in algebra can make operations, like using the distributive property, straightforward.When distributing multiplication across an expression, such as \[(x-6)4\]this involves multiplying the number outside the parentheses by each term inside. So, 4 is multiplied by both 'x' and '-6'.Each component gets multiplied:

• First, multiply \(4 \times x\), resulting in \(4x\)
• Then, multiply \(4 \times -6\), simplifying to \(-24\)

The result is the simplified expression:\[4x - 24\]Mastering multiplication in algebra is essential for simplifying expressions but also serves as a building block for more complex algebraic functions and equations.