Combining Like Terms A Step By Step Guide

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In the realm of algebra, simplifying expressions is a fundamental skill. One of the core techniques for simplification is combining like terms. This process involves identifying terms within an algebraic expression that share the same variable and exponent, and then adding or subtracting their coefficients. Mastering this skill is crucial for solving equations, simplifying complex expressions, and laying the groundwork for more advanced algebraic concepts.

Understanding the Basics of Like Terms

Before diving into the process, it's essential to understand what constitutes a "like term." Like terms are terms that have the same variable(s) raised to the same power(s). The coefficient, which is the numerical factor in front of the variable, can be different. For instance, 3x^2 and -5x^2 are like terms because they both have the variable x raised to the power of 2. However, 3x^2 and 3x are not like terms because the exponents are different, and 3x^2 and 3y^2 are not like terms because the variables are different. Similarly, 7xy and -2xy are like terms, while 7xy and 7x are not.

Think of it like this: you can only add or subtract things that are of the same "type." You can add apples to apples, but you can't directly add apples to oranges. In algebra, the variable part of a term defines its "type."

Steps to Combine Like Terms

The process of combining like terms is straightforward and can be broken down into a few key steps:

  1. Identify Like Terms: The first step is to carefully examine the expression and identify terms that have the same variable(s) and exponent(s). It can be helpful to use different shapes, colors, or underlines to group like terms together. For example, in the expression 5y^2 - 5y + 15 - 14y^2 - 5y + 7, we can identify 5y^2 and -14y^2 as like terms, -5y and -5y as like terms, and 15 and 7 as like terms (constant terms are always like terms).
  2. Rearrange Terms (Optional): While not always necessary, rearranging the terms so that like terms are next to each other can make the combining process clearer and reduce the chance of errors. This utilizes the commutative property of addition, which states that the order in which numbers are added does not affect the sum. For our example, we could rewrite the expression as 5y^2 - 14y^2 - 5y - 5y + 15 + 7.
  3. Combine Coefficients: Once you've identified and grouped the like terms, the next step is to add or subtract their coefficients. The coefficient is the number that multiplies the variable part of the term. For example, in the term 5y^2, the coefficient is 5. To combine like terms, simply add or subtract the coefficients and keep the variable part the same. Remember the rules for adding and subtracting integers. In our example, we have:
    • 5y^2 - 14y^2 = (5 - 14)y^2 = -9y^2
    • -5y - 5y = (-5 - 5)y = -10y
    • 15 + 7 = 22
  4. Write the Simplified Expression: After combining the coefficients of like terms, write the resulting expression. This will be the simplified form of the original expression. In our example, the simplified expression is -9y^2 - 10y + 22.

Example Problem: Step-by-Step Solution

Let's work through the example provided: 5y^2 - 5y + 15 - 14y^2 - 5y + 7

  1. Identify Like Terms:
    • 5y^2 and -14y^2 (terms with y^2)
    • -5y and -5y (terms with y)
    • 15 and 7 (constant terms)
  2. Rearrange Terms (Optional):
    • 5y^2 - 14y^2 - 5y - 5y + 15 + 7
  3. Combine Coefficients:
    • (5 - 14)y^2 = -9y^2
    • (-5 - 5)y = -10y
    • 15 + 7 = 22
  4. Write the Simplified Expression:
    • -9y^2 - 10y + 22

Therefore, 5y^2 - 5y + 15 - 14y^2 - 5y + 7 simplifies to -9y^2 - 10y + 22.

Common Mistakes to Avoid

When combining like terms, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them:

  • Combining Unlike Terms: This is the most frequent error. Remember, you can only combine terms that have the same variable(s) raised to the same power(s). Don't add or subtract terms like x^2 and x, or xy and x.
  • Incorrectly Adding/Subtracting Coefficients: Pay close attention to the signs (positive or negative) of the coefficients. Make sure you're applying the rules of integer arithmetic correctly. A common mistake is to forget the negative sign when subtracting a negative coefficient.
  • Forgetting to Include All Terms: Double-check that you've accounted for every term in the original expression. It's easy to accidentally overlook a term, especially in longer expressions.
  • Changing the Exponents: When combining like terms, you add or subtract the coefficients, but you do not change the exponents. The variable part of the term stays the same. For example, 3x^2 + 2x^2 = 5x^2, not 5x^4.

Advanced Examples and Applications

The principle of combining like terms extends to more complex expressions involving multiple variables and exponents. For example, consider the expression:

3a^2b - 4ab^2 + 5a^2b + 2ab - 7ab^2

  1. Identify Like Terms:
    • 3a^2b and 5a^2b
    • -4ab^2 and -7ab^2
    • 2ab (no other like term)
  2. Rearrange Terms (Optional):
    • 3a^2b + 5a^2b - 4ab^2 - 7ab^2 + 2ab
  3. Combine Coefficients:
    • (3 + 5)a^2b = 8a^2b
    • (-4 - 7)ab^2 = -11ab^2
    • 2ab (remains unchanged)
  4. Write the Simplified Expression:
    • 8a^2b - 11ab^2 + 2ab

Combining like terms is not just an algebraic exercise; it's a fundamental skill used in various mathematical contexts, including:

  • Solving Equations: Simplifying expressions by combining like terms is often a crucial step in solving algebraic equations.
  • Graphing Functions: When working with functions, you may need to simplify expressions before you can graph them.
  • Calculus: In calculus, simplifying expressions is essential for finding derivatives and integrals.
  • Real-World Applications: Many real-world problems can be modeled using algebraic expressions. Simplifying these expressions can make the problem easier to solve.

Practice Problems

To solidify your understanding of combining like terms, try these practice problems:

  1. Simplify: 7x - 3x + 2 + 5x - 8
  2. Simplify: 4y^2 + 2y - 6y^2 - y + 3
  3. Simplify: 9ab - 2a + 5ab + 3a - ab
  4. Simplify: 6p^2q - 3pq^2 + 2p^2q + 4pq^2 - pq

(Answers: 1. 9x - 6, 2. -2y^2 + y + 3, 3. 13ab + a, 4. 8p^2q + pq^2 - pq)

Conclusion

Combining like terms is a fundamental skill in algebra that allows you to simplify expressions and solve equations effectively. By understanding the concept of like terms, following the steps outlined in this guide, and avoiding common mistakes, you can master this skill and build a strong foundation for further algebraic studies. Remember, practice is key to proficiency, so work through plenty of examples and problems to solidify your understanding. Mastering the combination of like terms will serve you well in your mathematical journey.

By combining like terms, you transform a complex expression into a more manageable form, paving the way for solving equations and tackling more advanced mathematical challenges. This foundational skill is essential for success in algebra and beyond.