Combining like terms is a fundamental concept in algebra that simplifies expressions and makes them easier to work with. Understanding how to combine like terms is crucial for solving equations, simplifying expressions, and tackling more advanced algebraic concepts. This comprehensive guide will walk you through the process step-by-step, providing clear explanations and examples.
What are Like Terms?
Before we dive into combining like terms, let's define what they are. Like terms are terms that have the same variables raised to the same powers. The coefficients (the numbers in front of the variables) can be different, but the variables and their exponents must be identical.
Examples of Like Terms:
- 3x and 7x (same variable 'x', same exponent 1)
- 2y² and -5y² (same variable 'y', same exponent 2)
- 4ab and -ab (same variables 'a' and 'b', same exponents 1)
Examples of Unlike Terms:
- 2x and 2y (different variables)
- 3x² and 3x (same variable, different exponents)
- 5ab and 5a²b (same variables, different exponents)
Step-by-Step Guide to Combining Like Terms
Combining like terms involves adding or subtracting the coefficients of the like terms while keeping the variables and their exponents the same. Here's a step-by-step process:
Step 1: Identify Like Terms
Carefully examine the expression and identify all the like terms. It can be helpful to underline or circle like terms with the same color to visually group them.
Step 2: Group Like Terms
Rewrite the expression, grouping the like terms together. This makes the addition and subtraction process easier to manage. Remember to include the signs (+ or -) in front of each term.
Step 3: Combine the Coefficients
Add or subtract the coefficients of the grouped like terms. Remember that subtracting a number is the same as adding its opposite (e.g., 5 - 3 is the same as 5 + (-3)).
Step 4: Write the Simplified Expression
Write the simplified expression, combining the results from Step 3 with the corresponding variables and exponents.
Examples of Combining Like Terms
Let's work through some examples to solidify your understanding:
Example 1:
Simplify: 3x + 5y + 2x - y
- Identify Like Terms:
3xand2xare like terms;5yand-yare like terms. - Group Like Terms:
(3x + 2x) + (5y - y) - Combine Coefficients:
(3 + 2)x + (5 - 1)y=5x + 4y - Simplified Expression:
5x + 4y
Example 2:
Simplify: 2a² + 4ab - a² + 3ab
- Identify Like Terms:
2a²and-a²are like terms;4aband3abare like terms. - Group Like Terms:
(2a² - a²) + (4ab + 3ab) - Combine Coefficients:
(2 - 1)a² + (4 + 3)ab=a² + 7ab - Simplified Expression:
a² + 7ab
Example 3 (with more complex terms):
Simplify: 4x³y² - 2x²y³ + 3x³y² + 5x²y³
- Identify Like Terms:
4x³y²and3x³y²are like terms;-2x²y³and5x²y³are like terms. - Group Like Terms:
(4x³y² + 3x³y²) + (-2x²y³ + 5x²y³) - Combine Coefficients:
(4 + 3)x³y² + (-2 + 5)x²y³=7x³y² + 3x²y³ - Simplified Expression:
7x³y² + 3x²y³
Practice Makes Perfect!
The key to mastering combining like terms is practice. Work through numerous examples, gradually increasing the complexity of the expressions. You'll quickly become proficient in simplifying algebraic expressions and building a strong foundation in algebra. Remember to always double-check your work to ensure accuracy!
