Problem 42 - Introductory and Intermediate Algebra for College Students 4th

Question

Use a form of the distributive property to rewrite each algebraic expression without parentheses. $$(5 x+3) 6$$

Step-by-Step Solution

Answer

The simplified expression for \( (5x + 3)6 \) using the distributive property is \( 30x + 18 \)

Step 1: Identify the Terms

The expression \( (5x + 3)6 \) has two terms inside the parentheses: 5x and 3.

Step 2: Apply the Distributive Property

We multiply each term inside the parentheses by 6: \( 6 \times 5x \) and \( 6 \times 3 \)

Step 3: Simplify the Expression

Performing the multiplications gives: \( 6 * 5x = 30x \) and \( 6 * 3 = 18 \). Hence, the simplified expression is \( 30x + 18 \)

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is fundamental to solving many math problems. An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols, but no equals sign. For instance, in the expression \((5x+3)6\), the term \(5x\) is an algebraic expression within parentheses indicating multiplication with 6 outside. Here:\
\- **5** is a coefficient, a number that multiplies the variable.\
\- **x** is a variable representing an unknown number.\
\- **3** is a constant, a fixed value added to the term that makes the expression more than just a single term.\
These components come together to form a complete algebraic expression. It's important to identify and understand each part to handle operations like distribution correctly.

Simplification

Simplification involves reducing an expression to its simplest form. For the expression \((5x + 3)6\), simplifying means using the distributive property effectively. The goal is to make the expression easier to read and work with by removing the parentheses.\
\
Steps to simplify are:\
\- **Distribute**: Apply the distributive property. Multiply each term inside the parentheses by the number outside. So, you multiply \(5x\) by 6 and 3 by 6.\
\- **Calculate**: Perform the multiplications. Here, \(6 \times 5x = 30x\) and \(6 \times 3 = 18\).\
\
Once multiplication is done, combine like terms if any, but in this case, \(30x\) and \(18\) are unlike terms due to the presence of "x" only in one term. The final simplified expression becomes \(30x + 18\). Simplification makes expressions more straightforward and prepares them for further operations if needed.

Multiplication in Algebra

In algebra, multiplication extends the basic arithmetic operation to include variables. The expression \((5x + 3)6\) demonstrates how to apply multiplication in algebra by distributing the factor of 6 to both terms within parentheses.\
\
Key points about multiplication in algebra are:\
\- Multiplication with a variable follows the same principles as multiplying numbers. When multiplying by a constant, like multiplying \(5x\) by 6, the operation is carried out on the coefficient: \(5 \times 6 = 30\), resulting in \(30x\).\
\- The multiplication distributes over addition within the parentheses, enforcing the distributive property. This means both \(5x\) and \(3\) get multiplied by 6 separately.\
\
This approach helps to break down complex algebraic expressions, making them easier to manage and solve. Knowing how multiplication distributes across terms is crucial to grasping more intricate algebra concepts.