Combine Like Terms 3x 2x 3y-7y

Combining Like Terms: A Comprehensive Guide

Combining like terms is a fundamental concept in algebra, crucial for simplifying expressions and solving equations. This comprehensive guide will delve into the intricacies of combining like terms, providing a step-by-step approach, examples, and advanced applications to solidify your understanding. We'll specifically address the example expression, 3x + 2x + 3y - 7y, and expand upon the principles to handle more complex scenarios.

Understanding Like Terms

Before diving into the mechanics of combining like terms, it's essential to understand what constitutes a "like term." Like terms are terms that have the same variables raised to the same powers. Let's break this down:

• Variables: These are the letters (e.g., x, y, z) representing unknown quantities.
• Powers (Exponents): These indicate how many times a variable is multiplied by itself (e.g., x², y³, z). If a variable doesn't have a visible exponent, it's understood to be raised to the power of 1 (e.g., x = x¹).

Examples of Like Terms:

• 3x and 2x (same variable, x, raised to the power of 1)
• 5y² and -2y² (same variable, y, raised to the power of 2)
• -7ab and 4ab (same variables, a and b, each raised to the power of 1)

Examples of Unlike Terms:

• 3x and 3y (different variables)
• 2x² and 2x (same variable but different powers)
• 4ab and 4a²b (same variables but different powers)

Combining Like Terms: The Process

Combining like terms involves adding or subtracting the coefficients (the numbers in front of the variables) of the like terms. The variable part remains unchanged.

Let's illustrate this with our example expression: 3x + 2x + 3y - 7y

1. Identify Like Terms: We have two sets of like terms:

   • 3x and 2x
   • 3y and -7y

2. Combine Coefficients:

   • For the 'x' terms: 3 + 2 = 5. So, 3x + 2x simplifies to 5x.
   • For the 'y' terms: 3 + (-7) = -4. So, 3y - 7y simplifies to -4y.

3. Write the Simplified Expression: Combining the simplified terms, we get: 5x - 4y.

Therefore, 3x + 2x + 3y - 7y = 5x - 4y. This is the simplest form of the expression because we cannot combine 5x and -4y further; they are unlike terms.

More Complex Examples

Let's tackle more challenging examples to solidify your understanding:

Example 1:

4a²b + 2ab² - 3a²b + 5ab²

1. Identify Like Terms:

   • 4a²b and -3a²b
   • 2ab² and 5ab²

2. Combine Coefficients:

   • 4a²b - 3a²b = a²b
   • 2ab² + 5ab² = 7ab²

3. Simplified Expression: a²b + 7ab²

Example 2:

5x³ - 2x² + 7x - 3x³ + 4x² - 2x

1. Identify Like Terms:

   • 5x³ and -3x³
   • -2x² and 4x²
   • 7x and -2x

2. Combine Coefficients:

   • 5x³ - 3x³ = 2x³
   • -2x² + 4x² = 2x²
   • 7x - 2x = 5x

3. Simplified Expression: 2x³ + 2x² + 5x

Example 3: Incorporating Parentheses

2(x + 3y) - 4(2x - y)

1. Distribute: First, distribute the numbers outside the parentheses to the terms inside:

   • 2(x + 3y) = 2x + 6y
   • -4(2x - y) = -8x + 4y

2. Combine Like Terms:

   • 2x - 8x = -6x
   • 6y + 4y = 10y

3. Simplified Expression: -6x + 10y

Applications of Combining Like Terms

Combining like terms is not just an algebraic exercise; it's a crucial skill applied extensively in various mathematical and real-world contexts:

• Solving Equations: Simplifying equations by combining like terms makes them easier to solve.
• Geometry: Calculating perimeters and areas often involves combining like terms.
• Physics: Many physics formulas require manipulating algebraic expressions, including combining like terms.
• Data Analysis: Simplifying complex data representations frequently involves this process.

Common Mistakes to Avoid

While combining like terms seems straightforward, certain errors can occur:

• Combining Unlike Terms: This is the most common mistake. Remember, only like terms can be combined.
• Incorrectly Combining Coefficients: Pay close attention to the signs (+ or -) of the coefficients.
• Forgetting to Distribute: When dealing with parentheses, ensure that you correctly distribute the term outside the parentheses to every term inside.

Advanced Techniques and Further Exploration

Once you've mastered the basics, consider these advanced concepts:

• Polynomial Operations: Combining like terms is fundamental to adding, subtracting, multiplying, and dividing polynomials.
• Factoring: After combining like terms, you can often factor the simplified expression to further reduce it.
• Solving Systems of Equations: Combining like terms plays a vital role in solving systems of equations using methods like elimination.

Conclusion

Combining like terms is a fundamental algebraic skill that simplifies expressions, leading to more efficient problem-solving. By understanding like terms, mastering the combining process, and avoiding common mistakes, you'll build a strong foundation in algebra and its numerous applications. Remember to practice regularly with diverse examples to refine your skills and build confidence. This comprehensive guide provides a solid starting point for your journey into the world of algebraic simplification. Continue exploring advanced concepts to further enhance your mathematical proficiency.