Problem 93 - Elementary Algebra Within Reach

Question

Use the Distributive Property to expand the expression. $$4(2 x-5)$$

Step-by-Step Solution

Answer

The expanded form of the expression \(4(2x - 5)\) using the Distributive Property is \(8x - 20\).

Step 1: Understanding the Distributive Property

The Distributive Property states that for all real numbers \(a\), \(b\), and \(c\), the equation \(a(b + c) = ab + ac\) holds. In other words, multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products. This property can also be applied to subtraction.

Step 2: Applying the Distributive Property

Applying the distributive property to \(4(2x-5)\), we multiply \(4\) with each of the terms inside the parentheses. So, \(4\) is multiplied by \(2x\) and by \(-5\), respectively.

Step 3: Calculating the Product

Performing the multiplication yields: \(4*2x = 8x\) and \(4*-5 = -20\). Put them together, we have the expanded expression: \(8x-20\).

Key Concepts

Expanding Expressions

The process of expanding expressions involves using the distributive property to eliminate parentheses. By expanding, we transform an expression like \(4(2x - 5)\) into a simpler format that consists of individual terms separated by addition or subtraction. This makes it easier to work with algebraic equations and understand their components.

To expand, you follow these steps:

• Identify the term outside the parentheses, in this case, \(4\).
• Multiply this term by each term inside the parentheses, \(2x\) and \(-5\).

By performing these multiplications: \(4 \cdot 2x = 8x\) and \(4 \cdot (-5) = -20\), we derive the expanded expression: \(8x - 20\). This process reveals all components of the expression and is essential for simplifying and solving algebra problems.

Elementary Algebra

Elementary algebra is the foundation of all algebraic understanding. It deals with basic operations and the properties of numbers, variables, and expressions. At its core, elementary algebra involves learning about:

• Variables: Symbols like \(x\) that represent numbers.
• Expressions: Combinations of numbers, variables, and operations.
• Equations: Mathematical statements that show equality, usually containing one or more variables and requiring a solution.

The distributive property, which we are using in expanding expressions, is a key aspect of elementary algebra. It allows us to handle expressions involving parentheses effectively. It also illustrates how multiplication interacts with addition and subtraction.

Multiplication in Algebra

In algebra, multiplication extends beyond simple arithmetic. It involves variables and can interact with addition and subtraction through properties like distribution. Multiplying in algebra often integrates closely with the concept of the distributive property. Here’s how multiplication works when combined with other algebraic operations:

• Multiplying Variables: \( a \cdot a = a^2 \), meaning the variable is raised to the power of two.
• Using Coefficients: When multiplying terms with coefficients, multiply the numbers first. For example, \(4 \cdot 2x = 8x\).

When dealing with expressions like \(4(2x - 5)\), multiplication is used to "distribute" the \(4\) over both terms within the parentheses. This highlights the associative nature of multiplication, making it possible to simplify and reorganize complex expressions into manageable components.