Simplify By Combining Like Terms 5a+2b-3a+4

Simplifying Algebraic Expressions: A Comprehensive Guide to Combining Like Terms

Algebra, at its core, is about finding patterns and relationships between numbers and variables. One of the fundamental skills in algebra is simplifying expressions by combining like terms. This process streamlines complex expressions, making them easier to understand and manipulate. This comprehensive guide will delve into the intricacies of combining like terms, providing a step-by-step approach, illustrative examples, and practical applications.

Understanding Like Terms

Before diving into the process of simplification, it’s crucial to grasp the concept of "like terms." Like terms are terms in an algebraic expression that have the same variable(s) raised to the same power. Consider these examples:

• Like Terms: 3x and 7x (both have the variable 'x' raised to the power of 1)
• Like Terms: -2y² and 5y² (both have the variable 'y' raised to the power of 2)
• Like Terms: 4ab and -9ab (both have the variables 'a' and 'b', each raised to the power of 1)
• Unlike Terms: 2x and 2y (different variables)
• Unlike Terms: 3x² and 5x (different powers of the variable 'x')
• Unlike Terms: 6a and 6a²b (different variables and powers)

Identifying like terms is the first crucial step in simplifying expressions. Once identified, combining them becomes straightforward.

Combining Like Terms: A Step-by-Step Guide

The process of combining like terms involves adding or subtracting the coefficients (the numerical part of a term) of the like terms while keeping the variables and their exponents unchanged. Let’s break it down:

1. Identify Like Terms: Carefully examine the algebraic expression and identify all terms containing the same variable(s) raised to the same power. Circle or underline these terms to make identification easier, especially in more complex expressions.

2. Group Like Terms: Rewrite the expression, grouping the like terms together. This step is crucial for organizing the expression and avoiding errors. Use parentheses to group the terms if necessary, but remember that the sign preceding each term stays with the term.

3. Combine Coefficients: Add or subtract the coefficients of the like terms. Remember that the sign in front of a term is part of its coefficient. For example, -5x means -5 is the coefficient.

4. Write the Simplified Expression: Write down the simplified expression, incorporating the results from Step 3. The variables and their powers remain unchanged.

Example: Simplifying 5a + 2b - 3a + 4

Let's apply this step-by-step guide to the expression you provided: 5a + 2b - 3a + 4.

Step 1: Identify Like Terms:

• The like terms are 5a and -3a (both have the variable 'a' raised to the power of 1).
• The term 2b is a unique term.
• The term 4 is a constant term (a term without a variable).

Step 2: Group Like Terms:

(5a - 3a) + 2b + 4

Step 3: Combine Coefficients:

5a - 3a = 2a

Step 4: Write the Simplified Expression:

The simplified expression is 2a + 2b + 4. We cannot simplify this expression further because the remaining terms are unlike terms.

More Complex Examples: Mastering the Technique

Let's tackle more complex expressions to solidify your understanding:

Example 1: 3x² + 5x - 2x² + 7x - 8

Step 1: Identify Like Terms:

• Like terms: 3x² and -2x²
• Like terms: 5x and 7x
• Constant term: -8

Step 2: Group Like Terms:

(3x² - 2x²) + (5x + 7x) - 8

Step 3: Combine Coefficients:

3x² - 2x² = x² 5x + 7x = 12x

Step 4: Write the Simplified Expression:

The simplified expression is x² + 12x - 8

Example 2: 4xy + 2x - 6xy + 3y + 5x

Step 1: Identify Like Terms:

• Like terms: 4xy and -6xy
• Like terms: 2x and 5x
• Unique term: 3y

Step 2: Group Like Terms:

(4xy - 6xy) + (2x + 5x) + 3y

Step 3: Combine Coefficients:

4xy - 6xy = -2xy 2x + 5x = 7x

Step 4: Write the Simplified Expression:

The simplified expression is -2xy + 7x + 3y

Example 3: 2a²b + 3ab² - 5a²b + 7ab² + a²b

Step 1: Identify Like Terms:

• Like terms: 2a²b, -5a²b, a²b
• Like terms: 3ab², 7ab²

Step 2: Group Like Terms:

(2a²b - 5a²b + a²b) + (3ab² + 7ab²)

Step 3: Combine Coefficients:

2a²b - 5a²b + a²b = -2a²b 3ab² + 7ab² = 10ab²

Step 4: Write the Simplified Expression:

The simplified expression is -2a²b + 10ab²

Practical Applications and Importance

The ability to simplify algebraic expressions by combining like terms is crucial for various aspects of mathematics and its applications:

• Solving Equations: Simplifying expressions is a prerequisite for solving equations. Simplifying allows you to isolate the variable and find its value.

• Graphing Functions: Simplified expressions make it easier to graph functions. A simplified form allows for a clearer understanding of the function's behavior and characteristics.

• Real-World Problem Solving: Many real-world problems can be modeled using algebraic expressions. Simplifying these expressions is crucial for obtaining meaningful solutions. For instance, in calculating the total cost of items in a shopping cart or determining profit in a business, combining like terms simplifies the calculation.

• Advanced Algebraic Concepts: Mastering the ability to combine like terms provides a strong foundation for more advanced concepts in algebra, such as factoring, expanding expressions, and solving polynomial equations.

Common Mistakes to Avoid

• Ignoring Signs: Pay close attention to the sign (positive or negative) in front of each term. The sign is a part of the coefficient and influences the result of combining like terms.

• Incorrectly Combining Unlike Terms: Remember that you can only combine terms with the same variable(s) raised to the same power.

• Errors in Arithmetic: Double-check your arithmetic during the coefficient combination process. A simple arithmetic error can lead to an incorrect simplified expression.

• Not Grouping Like Terms: Always group like terms before combining them. This helps you organize the expression and avoid mistakes.

Conclusion: Mastering the Fundamentals

Combining like terms is a cornerstone skill in algebra, crucial for simplifying expressions and progressing to more advanced topics. By diligently following the step-by-step guide, working through numerous examples, and being aware of common mistakes, you can master this fundamental concept and build a strong foundation in algebra. With practice, this seemingly simple process will become second nature, greatly enhancing your understanding and abilities in mathematics and its various applications.