Problem 35 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each expression without parentheses. Then simplify the result. See Example 4. \(2(3 x+5)\)

Step-by-Step Solution

Answer

The expression simplifies to \(6x + 10\).

Step 1: Apply the Distributive Property

The distributive property states that for any numbers \(a\), \(b\), and \(c\), the expression \(a(b+c)\) can be expanded to \(ab + ac\). Apply this property to \(2(3x + 5)\). This gives us \(2 \cdot 3x + 2 \cdot 5\).

Step 2: Simplify the Individual Terms

Simplify each term calculated in the previous step. For \(2 \cdot 3x\), simplify to \(6x\). For \(2 \cdot 5\), simplify to \(10\).

Step 3: Combine the Simplified Terms

Combine them back into an expression: \(6x + 10\). This is the simplified version of the original expression after distributing and combining terms.

Key Concepts

Algebra

Algebra is the branch of mathematics that uses symbols and letters to represent numbers and quantities in equations and expressions. This makes it possible to work with unknown values and solve problems in a general way. In algebra, we often encounter expressions that include variables like \(x\), which symbolize numbers that we want to find out or manipulate.

In the exercise provided, we're using algebra to simplify an expression by applying a property known as the distributive property. Algebraic expressions can become quite complex, but breaking them down using these kinds of properties helps us understand and simplify them. Understanding basic operators and algebraic properties is crucial for effectively manipulating and solving equations in algebra.

Simplifying Expressions

Simplifying expressions is a critical skill in algebra that involves transforming a complex expression into a simpler form while retaining the same value. This process can involve a variety of techniques such as combining like terms, factoring, and using mathematical properties.

In our exercise, we focused on removing parentheses from the expression \(2(3x + 5)\) by distributing the multiplication, a step which leads us to a simpler form: \(6x + 10\).

**Key Steps in Simplifying Expressions**:

• Identify terms and coefficients, which are the numbers in front of the variables.
• Apply the distributive property if needed to remove parentheses.
• Combine like terms, which are terms that have identical variable parts.
• Simplify numerical values for easy readability.

Simplifying makes it easier to evaluate expressions and aids in comparisons or expansions needed in solving equations or inequalities.

Mathematical Properties

Mathematical properties are rules that apply to numbers and operations. They provide helpful strategies for solving expressions and equations. These include the distributive property, associative property, and commutative property, among others.

The distributive property, which is central to our exercise, states that multiplying a number by a group of numbers added together is the same as doing each multiplication separately. Mathematically, this is expressed as \(a(b + c) = ab + ac\).

**Importance of Mathematical Properties**:

• Help in transforming and simplifying complex algebraic expressions.
• Allow for the verification and validation of solutions.
• Provide a foundation for more advanced mathematical concepts.

By using these properties, we can greatly simplify the problem-solving process in algebra, allowing for more efficient and effective solutions.