Problem 17 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$ 3(x+4) $$

Step-by-Step Solution

Answer

The expression without parentheses is \(3x + 12\).

Step 1: Apply the Distributive Property

Apply the distributive property to the expression \(3(x+4)\), by multiplying each term inside the parentheses by the number outside, which is 3 here. So the expression becomes \(3 \cdot x + 3 \cdot 4\).

Step 2: Simplify the Expression

Do the multiplication \(3 \cdot x\) to be \(3x\) and \(3 \cdot 4\) to be 12. Therefore, the expression simplifies to \(3x + 12\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. In the expression \(3(x+4)\), you can identify several components: the number 3, the variable \(x\), and the operation within the parentheses. These expressions may look complex at first, but once you understand how to manage each element, they become less intimidating.

Here are the main parts of an algebraic expression:

• **Coefficients**: The numbers multiplying the variables, like the 3 in \(3x\).
• **Variables**: Symbols like \(x\), representing unknown numbers.
• **Constants**: Regular numbers such as the 4 in the expression, which do not change.
• **Operators**: Symbols like + and - that show operations to perform.

Grasping these basics will help you manipulate and simplify algebraic expressions with confidence.

Simplification

Simplification is the process of making an expression easier to understand and work with, by performing operations to reduce it to its simplest form. In this context, it involves using operations like addition, subtraction, and multiplication.

When simplifying expressions like \(3(x+4)\), your goal is to combine like terms and remove any unnecessary parentheses. This often makes the expression more intuitive and easier to work with when solving more complex equations.

To simplify properly, you should always:

• Follow the order of operations, often remembered by PEMDAS/BODMAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Perform multiplication first, as seen in \(3 \cdot (x + 4)\).
• Combine like terms when possible to tidy up the expression, like \(3x + 12\).

After simplification, the expression is in a clearer form, making it straightforward to identify individual elements and their relationships.

Multiplication in Algebra

Multiplication in algebra can initially seem more complex than basic arithmetic, as it incorporates variables. However, it follows similar principles. Multiplication is about scaling numbers or terms, which can include both numbers and variables.

Consider the expression \(3(x + 4)\). By multiplying 3, the term outside the parentheses, with each individual term inside the parentheses, the expression gets expanded. This follows the Distributive Property, which indicates that \(a(b + c) = ab + ac\).

Steps to multiply in algebra:

• **Identify terms** to be distributed and multiplied: 3 is multiplied by \(x\) and 4.
• **Perform multiplication** for each term separately: \(3 \cdot x = 3x\) and \(3 \cdot 4 = 12\).
• **Combine results** to form the simplified expression: \(3x + 12\).

This approach not only simplifies the expression but also demonstrates how multiplication links different parts of an algebraic equation. Understanding this concept is pivotal as algebra often requires simplifying and manipulating such expressions.