How To Combine Like Terms A Step-by-Step Guide

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In mathematics, simplifying expressions is a fundamental skill. One crucial technique is combining like terms. This involves identifying terms within an algebraic expression that share the same variable raised to the same power and then combining their coefficients. This process streamlines expressions, making them easier to understand and manipulate. In this comprehensive guide, we will delve into the concept of combining like terms, provide a step-by-step approach, and illustrate the process with examples.

Understanding Like Terms

Before we can effectively combine like terms, it is crucial to define what they are. Like terms are terms that have the same variable(s) raised to the same power(s). The coefficient, which is the numerical factor of the term, can be different. For example, in the expression 3x + 5x - 2y + 7y, the terms 3x and 5x are like terms because they both have the variable x raised to the power of 1. Similarly, -2y and 7y are like terms because they both have the variable y raised to the power of 1. However, 3x and -2y are not like terms because they have different variables.

To further illustrate this, consider the terms 4x^2, -7x^2, and 9x^2. These are all like terms because they have the same variable, x, raised to the same power, 2. On the other hand, 4x^2 and 4x are not like terms because, although they share the same variable x, the powers are different (2 versus 1). Similarly, 5xy and -2xy are like terms, but 5xy and 5x are not because the former includes both variables x and y while the latter only includes x.

Identifying Like Terms

The first step in combining like terms is to correctly identify them within an expression. This involves carefully examining each term and comparing their variable parts. Pay close attention to both the variables and their exponents. A systematic approach can help prevent errors.

Consider the expression 7a - 3b + 2a^2 - 5a + 8b - a^2. To identify the like terms, we can group them as follows:

  • Terms with a: 7a and -5a
  • Terms with b: -3b and 8b
  • Terms with a^2: 2a^2 and -a^2

This grouping makes it clear which terms can be combined. Mistaking terms with different powers or variables can lead to incorrect simplification. For instance, in the expression 4x^3 + 2x^2 - x^3 + 5x, it is essential to recognize that 4x^3 and -x^3 are like terms, while 2x^2 and 5x are distinct and cannot be combined with the cubic terms.

Step-by-Step Process for Combining Like Terms

Once you have identified the like terms, you can proceed to combine them. This involves adding or subtracting the coefficients of the like terms while keeping the variable part the same. Here is a step-by-step process:

  1. Identify Like Terms: As discussed earlier, group terms with the same variable(s) raised to the same power(s).
  2. Rearrange the Expression (Optional): Sometimes, rearranging the expression to group like terms together can make the process clearer. This is especially helpful for complex expressions. For example, you can rewrite 5x + 3y - 2x + y as 5x - 2x + 3y + y.
  3. Combine Coefficients: Add or subtract the coefficients of the like terms. Remember to pay attention to the signs (positive or negative) of the coefficients.
  4. Write the Simplified Expression: Write the new term with the combined coefficient and the original variable part.

For example, consider the expression 8x + 4x - 3x. The terms 8x, 4x, and -3x are like terms. Combining their coefficients, we have 8 + 4 - 3 = 9. Therefore, the simplified expression is 9x.

Detailed Examples

Let’s walk through a few more examples to illustrate the process:

Example 1: Simplify the expression 6y - 2y + 9y - 4y.

  1. Identify Like Terms: All terms have the variable y raised to the power of 1, so they are all like terms.
  2. Combine Coefficients: 6 - 2 + 9 - 4 = 9
  3. Write the Simplified Expression: 9y

Example 2: Simplify the expression 5a^2 - 2a + 3a^2 + 7a - a^2.

  1. Identify Like Terms: The like terms are 5a^2, 3a^2, and -a^2 (for terms with a^2), and -2a and 7a (for terms with a).
  2. Rearrange the Expression: 5a^2 + 3a^2 - a^2 - 2a + 7a
  3. Combine Coefficients: For a^2 terms: 5 + 3 - 1 = 7. For a terms: -2 + 7 = 5
  4. Write the Simplified Expression: 7a^2 + 5a

Example 3: Simplify the expression 4x^2y - 3xy + 2x^2y + 5xy - xy^2.

  1. Identify Like Terms: The like terms are 4x^2y and 2x^2y (for terms with x^2y), and -3xy and 5xy (for terms with xy). The term -xy^2 is unique and cannot be combined with others.
  2. Rearrange the Expression: 4x^2y + 2x^2y - 3xy + 5xy - xy^2
  3. Combine Coefficients: For x^2y terms: 4 + 2 = 6. For xy terms: -3 + 5 = 2
  4. Write the Simplified Expression: 6x^2y + 2xy - xy^2

Common Mistakes to Avoid

Combining like terms is a straightforward process, but several common mistakes can lead to incorrect simplifications. Being aware of these pitfalls can help prevent errors:

  • Combining Unlike Terms: This is the most common mistake. Ensure that you only combine terms with the same variable(s) raised to the same power(s). For example, do not combine 3x^2 and 4x.
  • Ignoring Signs: Pay close attention to the signs (positive or negative) of the coefficients. A negative sign in front of a term must be included when combining.
  • Incorrect Arithmetic: Double-check your arithmetic when adding or subtracting coefficients. Simple calculation errors can lead to incorrect results.
  • Forgetting to Distribute: If the expression involves parentheses, remember to distribute any coefficients or negative signs properly before combining like terms. For example, in 2(x + 3) - 4x, you need to distribute the 2 before combining like terms.

Applying Combining Like Terms

The ability to combine like terms is essential in various mathematical contexts, including:

  • Simplifying Algebraic Expressions: As demonstrated, combining like terms reduces complex expressions to simpler forms, making them easier to work with.
  • Solving Equations: When solving equations, combining like terms can simplify both sides, making the equation easier to solve.
  • Factoring Polynomials: Factoring often involves identifying and combining like terms within a polynomial.
  • Calculus: Simplifying expressions by combining like terms is a preliminary step in many calculus problems.

Practice Problems

To solidify your understanding, try simplifying the following expressions:

  1. 9m - 4m + 6m - m
  2. 2p^2 + 5p - p^2 - 3p + 4
  3. 7ab - 3a + 2ab + 5a - b
  4. 3(x - 2) + 5x - 1

Real-World Applications

While combining like terms is a fundamental algebraic skill, it has real-world applications in various fields. For example, in engineering, simplifying expressions involving forces, velocities, and other physical quantities often requires combining like terms. In finance, calculating total costs or profits may involve combining like terms in algebraic expressions. In computer science, simplifying Boolean expressions uses similar principles to optimize code and circuit designs.

Solving the Initial Problem: Combining Like Terms with Fractions

Now, let's address the initial problem presented: Simplify the expression:

(1/7)x - (3/7)y - (3/5)x + (6/17)y

This expression involves fractions, which adds a layer of complexity, but the same principles apply. We will combine the x terms and the y terms separately.

Step 1: Identify Like Terms

  • x terms: (1/7)x and -(3/5)x
  • y terms: -(3/7)y and (6/17)y

Step 2: Combine the x Terms

To combine (1/7)x and -(3/5)x, we need to find a common denominator for the fractions 1/7 and 3/5. The least common multiple (LCM) of 7 and 5 is 35. Convert the fractions:

  • (1/7)x = (5/35)x
  • -(3/5)x = -(21/35)x

Now, combine the x terms:

(5/35)x - (21/35)x = (5 - 21)/35 x = (-16/35)x

Step 3: Combine the y Terms

To combine -(3/7)y and (6/17)y, we need to find a common denominator for the fractions 3/7 and 6/17. The LCM of 7 and 17 is 119. Convert the fractions:

  • -(3/7)y = -(51/119)y
  • (6/17)y = (42/119)y

Now, combine the y terms:

-(51/119)y + (42/119)y = (-51 + 42)/119 y = (-9/119)y

Step 4: Write the Simplified Expression

Combine the simplified x and y terms:

(-16/35)x + (-9/119)y

So, the simplified expression is:

(-16/35)x - (9/119)y

Conclusion

Combining like terms is a fundamental skill in algebra that simplifies expressions and makes them easier to manipulate. By understanding what like terms are, following a systematic process, and avoiding common mistakes, you can confidently simplify algebraic expressions. This skill is essential for success in higher-level mathematics and has practical applications in various real-world fields. Mastering this technique will not only improve your algebraic abilities but also enhance your problem-solving skills in diverse contexts. Remember to practice regularly, and you will become proficient at combining like terms, making your mathematical journey smoother and more rewarding. Simplifying expressions by combining like terms is a core concept in algebra, offering a streamlined approach to handling mathematical problems across various domains. Whether it's for solving equations, tackling calculus, or applying mathematical principles in real-world scenarios, the ability to combine like terms proficiently is invaluable. This guide provides a thorough understanding and practical steps to master this skill, ensuring a solid foundation in algebraic manipulations.