Problem 79 - Beginning and Intermediate Algebra

Question

Use the distributive property to rewrite each expression. $$ 2(6 x+5) $$

Step-by-Step Solution

Answer

12x + 10

Step 1 - Identify the expression

The given expression is 2(6x + 5).

Step 2 - Apply the distributive property

The distributive property states that a(b + c) = ab + ac. In this case, a = 2, b = 6x, and c = 5. Apply the distributive property: 2(6x + 5) = 2 * 6x + 2 * 5.

Step 3 - Multiply the terms

Multiply each term: 2 * 6x = 12x and 2 * 5 = 10.

Step 4 - Combine the terms

Combine the results to get the final expression: 12x + 10.

Key Concepts

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operators. Think of them as mathematical phrases that represent quantities and their relationships. In our example, the expression is 2(6x + 5). Here, '6x' is a term involving a variable 'x' and '5' is a constant term. Such expressions appear frequently in algebra, and learning to manipulate them is crucial. Working with algebraic expressions helps us solve real-world problems by providing a structured way to handle quantities that vary.

Multiplication

Multiplication is one of the basic operations in mathematics. When dealing with algebraic expressions, multiplication often involves combining coefficients (numeric factors) with variables. For instance, in our exercise, we have to multiply 2 by each term inside the parentheses.
Following the distributive property, we multiply `2` by `6x` and `5`. Here’s how you can do it step-by-step:

• Multiply 2 by 6x: 2 * 6x = 12x
• Multiply 2 by 5: 2 * 5 = 10

Breaking down the multiplication step helps in understanding how each term is processed independently.

Basic Algebra

Basic algebra involves the manipulation of algebraic expressions to simplify them or solve for variables. The distributive property is a fundamental concept in algebra that allows us to distribute a multiplier across terms within parentheses.
In our example, using the distributive property: 2(6x + 5)
We get: 2 * 6x + 2 * 5, which simplifies to 12x + 10.
This property helps simplify and manage expressions effectively. Understanding and applying the distributive property is essential in algebra as it lays the foundation for more complex equations and operations.