Problem 35 - Algebra A Combined Function

Question

Use the distributive property to write each expression without parentheses. Then simplify the result, if possible. See Examples 7 through 12. $$ 2(3 x+5) $$

Step-by-Step Solution

Answer

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

Step 1: Apply the Distributive Property

To apply the distributive property, you need to multiply the term outside the parentheses by each term inside the parentheses. The expression is \(2(3x + 5)\). Multiply \(2\) by \(3x\) and \(2\) by \(5\). This gives: \(2 \times 3x + 2 \times 5\).

Step 2: Simplify Each Term

Now calculate each term separately. \(2 \times 3x = 6x\), and \(2 \times 5 = 10\).

Step 3: Combine the Terms

Add the simplified terms together to obtain the final expression. From Step 2, we determined the terms to be \(6x\) and \(10\). Therefore, the expression is \(6x + 10\).

Key Concepts

Simplification

In algebra, simplification refers to the process of reducing an expression to its most straightforward form. It involves performing all possible operations to create a simpler, yet equivalent, version of the original expression. Simplification enhances our understanding of the expression by removing complexities and redundancies.

The simplification process often involves combining like terms, which means adding or subtracting terms that have the same variable raised to the same power. For example, if you have the expression \( 3x + 4x \), you can simplify it to \( 7x \) because both terms contain the variable \( x \) to the power of one.

Another common step in simplification is reducing fractions to their lowest terms, which ensures there are no common factors between the numerator and the denominator other than one. Furthermore, simplification is essential because it provides a clearer view, making it easier to solve equations or to understand their properties more comprehensively.

• Combining like terms
• Reducing fractions
• Applying arithmetic operations

Algebraic Expressions

Algebraic expressions are a central concept in algebra, consisting of numbers, variables, and operations such as addition, subtraction, multiplication, and division. These expressions can range from simple forms like \( 2x + 5 \) to more complex formulas involving multiple variables and operations.

The primary purpose of algebraic expressions is to express mathematical ideas or relationships abstractly. They provide a way to model real-world situations and solve problems through the manipulation of symbols rather than specific numbers. In the expression used in this exercise, \( 2(3x + 5) \), we see a multiplication operation applied to a sum within parentheses. This is typical in algebraic manipulation where operations are grouped to apply certain properties more efficiently.

Algebraic expressions allow flexibility and generalization. Instead of dealing with specific numbers, algebraic expressions allow us to work with formulas and see patterns. This makes them valuable in developing problem-solving skills and understanding how various components relate to each other.

• Variables signify unknown or varying quantities
• Operations signify the interactions between those quantities
• alert the presence of more complex relationships

Multiplication

Multiplication is a fundamental operation in mathematics and is heavily utilized in algebra, especially in expressions and equations. It is the process of adding a number to itself a specified number of times. In algebra, multiplication often involves numbers, variables, or a combination of both.

When dealing with algebraic expressions, multiplication is used not only between numbers but also to spread a term across others within parentheses, applying the distributive property. For instance, to simplify \( 2(3x + 5) \), we multiply each term inside the parentheses by the number outside, \( 2 \). This distributes the multiplication over addition: \( 2 \times 3x + 2 \times 5 \).

Understanding multiplication in this context is crucial because it lays the groundwork for more advanced algebra topics like factoring or expanding expressions. It also ensures that algebraic transformations maintain equality and correctness.

• Spreads terms across expressions
• Frequently involves the distributive property
• Builds the basis for complex algebraic operations