Simplifying expressions is a fundamental technique in algebra that involves combining like terms to reduce the complexity of expressions. This process not only makes algebraic expressions easier to understand but also prepares them for further operations such as solving equations.

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Introduction

In this section, we explore how to simplify algebraic expressions by collecting like terms. Simplifying expressions is crucial for solving algebraic problems efficiently and forms the basis for more complex algebraic manipulations.

Identifying Like Terms

Understanding like terms is pivotal. They have the same variable(s) and powers. For instance, $2x$ and $5x$ are like terms because both involve the variable $x$ to the first power.

Examples of Like Terms

• $3y$ and $7y$
• $4a^2$ and $-2a^2$
• $5bc$ and $3bc$

Examples of Unlike Terms

• $3x$ and $3x^2$
• $4ab$ and $4a$
• $6y$ and $6z$

Techniques for Simplification

The primary steps in simplification include identifying, combining like terms, and possibly rearranging the expression for clarity.

Worked Examples

Example 1: Simplifying a Basic Expression

Simplify $2x + 3x$.

Solution:

Combine like terms: $2x + 3x$.

Final Answer: $5x$.

Example 2: A More Complex Expression

Simplify $4y - 2y + 3y^2 - y^2 + 6$.

Solution:

Combine like terms in steps:

$4y - 2y$

$3y^2 - y^2$

Calculation:

$4y - 2y = 2y$

$3y^2 - y^2 = 2y^2$

$6$ remains as is.

Final Answer: $2y^2 + 2y + 6$

Example 3: Simplification with Different Variables

Simplify $3a + 4b - a + 2b - 3$.

Solution:

Combine like terms:

$3a - a$

$4b + 2b$

Calculation:

$3a - a = 2a$

$4b + 2b = 6b$

$-3$ is constant.

Final Answer: $2a + 6b - 3$

Practice Problems

1. Simplify $7m + 3n - 2m + 5$.

Solution:

2. Simplify $5x^2 + 3 - 2x^2 + 4x - x$.

Solution:

3. Simplify $2a^3 - 3a^2 + 4a^3 + a^2$.

Solution:

Key Points to Remember

• Like terms share the same variables and exponents.
• Combine like terms by adding or subtracting their coefficients.
• Simplification can involve reorganizing the expression for clarity, but it's not mandatory.