Problem 4 - Algebra 1: Concepts and Skills

Question

Identify the like terms in the expression. \(4 x+2(x+1)\)

Step-by-Step Solution

Answer

The like terms in the given expression are \(4x\) and \(2x\).

Step 1: Simplify the expression

First distribute the multiplication of 2 to both terms inside the bracket. Hence we get \(4x + 2x + 2\).

Step 2: Identify like terms

The like terms here are the terms with the same variable 'x'. So, in this case, these are \(4x\) and \(2x\).

Key Concepts

Expression Simplification

Expression simplification involves reducing an algebraic expression to its simplest form. This means combining like terms and performing arithmetic operations. In this exercise, we begin with the expression \(4x + 2(x + 1)\). To simplify, we distribute and then combine.
- **Distribute First:** Apply the distributive property to eliminate brackets. Here, the term \(2(x + 1)\) requires distribution, leading to \(2 \times x + 2 \times 1\).
- **Combine Like Terms:** After distribution, simplify further by combining terms with the same variables. In this exercise, you combine \(4x\) and \(2x\) to get \(6x\).
The final result after simplification is \(6x + 2\). This approach helps in manageable and clear representations, essential for solving complex algebraic problems.

Distributive Property

The distributive property is vital in algebra, allowing the multiplication of a single term across terms within a parenthesis. With the expression \(2(x+1)\), apply this property by multiplying \(2\) with each term inside:
- **Steps of Application:** Distribute the outside term over each inside term: \(2 \cdot x\) gives \(2x\), and \(2 \cdot 1\) gives \(2\).
- **Purpose:** This simplifies expressions, removing parentheses and combining terms.
The property essentially transforms \(a(b + c)\) to \(ab + ac\). Practicing this ensures efficiency in algebraic calculations, aiding in solving equations or simplifying expressions.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operators forming a mathematical statement. In our expression \(4x + 2(x+1)\), we see a mix of constant numbers, variables, and operations. Misunderstanding these can complicate simplification.
- **Variables:** Symbols like \(x\) which represent unknown values.
- **Terms:** Parts separated by \(+\) or \(-\), such as \(4x\) or \(2\) in this expression.
- **Importance of Like Terms:** Useful in simplification as they have identical variable portions, enabling them to be combined.
Understanding these components is crucial to unpacking and working with algebraic expressions effectively. By knowing how to manipulate and simplify, students can tackle more complex algebraic problems with confidence.