Problem 38 - Introductory and Intermediate Algebra for College Students 4th

Question

Use a form of the distributive property to rewrite each algebraic expression without parentheses. $$6(3 x-2)$$

Step-by-Step Solution

Answer

The expression without parentheses is \(18x - 12\).

Step 1: Distribute the Multiplication

We apply the distributive property by multiplying 6 by both terms inside the brackets. The equation now changes to \(6 \times 3x - 6 \times 2\).

Step 2: Carry Out the Multiplication

Perform the multiplication operation. So, our equation becomes \(18x - 12\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In algebra, variables are symbols like \(x\), \(y\), or \(z\) that represent unknown values and can change within the context of mathematical problems.

For example, \(3x - 2\) is an algebraic expression. It consists of the variable \(x\), numbers 3 and 2, and an arithmetic operation (subtraction). Algebraic expressions can describe relationships, patterns, or general mathematical rules, and grasping their structure is essential in both elementary and advanced algebra.

Multiplication

Multiplication, one of the basic operations of arithmetic, involves calculating the total of one number (the multiplicand) when it is increased by the value of another number (the multiplier). In algebra, multiplication is used to combine constants with variables, to expand expressions, and to solve equations.

For instance, the multiplication of 6 by \(3x\) yields \(18x\), the product. The number 6 is the multiplier, and \(3x\) is the multiplicand. This operation is fundamental in simplifying algebraic expressions and appears in various forms, such as coefficients being multiplied by variables.

Simplifying Expressions

Simplifying expressions is a process of transforming a complex and lengthy expression into a simpler form without changing its value. The goal is to make the expression easier to understand and work with by combining like terms and using properties of arithmetic operations.

In our example, simplifying \(6(3x - 2)\) involves using the distributive property to eliminate the parentheses and reduce the expression to its simplest form. Starting with distribution, you multiply the number outside the parentheses, in this case, 6, by each term inside, giving us \(18x - 12\). The expression then becomes free of the parentheses and is considered simplified. This method aids in further problem-solving, such as solving equations or evaluating the expression for given values of variables.