Problem 38 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each expression without parentheses. Then simplify the result. See Example 4 . \(3(8 x-1)\)

Step-by-Step Solution

Answer

The expression simplifies to \( 24x - 3 \).

Step 1: Apply the Distributive Property

The distributive property states that for any numbers \( a \), \( b \), and \( c \), the expression \( a(b + c) \) is equal to \( ab + ac \). In this case, we'll apply the distributive property to \( 3(8x - 1) \). This is done by multiplying \( 3 \) by each term inside the parentheses: \( 3 \times 8x \) and \( 3 \times (-1) \).

Step 2: Perform the Multiplications

Calculate \( 3 \times 8x = 24x \) and \( 3 \times (-1) = -3 \). So, applying the distributive property, \( 3(8x - 1) = 24x - 3 \).

Step 3: Simplify the Expression

The result of the distributed expression is \( 24x - 3 \). There are no like terms to combine, so the expression is already simplified.

Key Concepts

Algebraic Expressions

Algebraic expressions are like containers that hold numbers, variables, and operations. They are integral to algebra, as they allow you to represent mathematical situations in general terms. For instance, the expression \( 3(8x - 1) \) consists of a number, variable, and operations like multiplication and subtraction. These expressions are the building blocks you'll use to solve equations and understand relationships between numbers.

Algebraic expressions can include:

• Constants: Numbers on their own, like \( 3 \).
• Variables: Symbols, usually letters, that represent unknown or changeable numbers, like \( x \).
• Operations: Mathematical processes such as addition (+), subtraction (-), multiplication (×), and division (÷).

Understanding how to manipulate and simplify these expressions is a crucial part of algebra.

Simplifying Expressions

Simplifying expressions means making them as straightforward as possible. This involves combining like terms and using properties like the distributive property to rewrite expressions in a simpler form.

In the exercise, we used the distributive property on \( 3(8x - 1) \) to eliminate the parentheses and simplify the expression to \( 24x - 3 \). Simplification requires identifying terms with the same variable and combining them, or rewriting expressions using standard algebraic rules. However, in the expression \( 24x - 3 \), there are no like terms to combine, which means it's already in its simplest form.

The benefits of simplifying expressions include:

• Making expressions easier to work with or solve.
• Helping to identify equivalent expressions.
• Simplifying calculations or comparisons in real-world problems.

In the journey through algebra, simplifying expressions plays a key role in achieving clarity and understanding.

Mathematical Operations

Mathematical operations are the processes you perform on numbers or symbols to obtain a result. They are the fundamental actions that drive calculations and manipulations in mathematics. Typically, these operations include addition, subtraction, multiplication, and division.

In algebra, understanding operations is crucial for applying properties like the distributive property. When we encounter \( 3(8x - 1) \), the operation of multiplication is used to distribute \( 3 \) to both terms inside the parentheses: \( 3 \times 8x \) and \( 3 \times (-1) \).

Key points about mathematical operations:

• They allow you to derive new expressions or results from existing ones.
• Order of operations (PEMDAS/BODMAS) dictates which operations to perform first.
• Properties like commutative, associative, and distributive simplify calculations.

By mastering these operations, you can solve complex equations and understand the underlying patterns in algebraic problems.