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 walk you through the process, explaining the underlying principles and providing numerous examples to solidify your understanding. We'll specifically address the expression "3x + 2x + 3y - 7y" and then expand to more complex scenarios.

What are Like Terms?

Before we dive into combining, let's define what "like terms" are. Like terms are terms in an algebraic expression that have the same variables raised to the same powers. This might seem confusing at first, but let's break it down:

• Variables: These are the letters in an algebraic expression (e.g., x, y, z).
• Powers (Exponents): These are the small numbers written slightly above and to the right of a variable (e.g., x², y³). If there's no exponent written, it's implicitly 1 (e.g., x = x¹).

Examples of Like Terms:

• 3x and 2x: Both have the variable 'x' raised to the power of 1.
• 5y² and -2y²: Both have the variable 'y' raised to the power of 2.
• -7 and 4: These are constants (numbers without variables) and are considered like terms.

Examples of Unlike Terms:

• 2x and 2y: Different variables.
• 4x² and 4x: Same variable but different powers.
• 6x and 6xy: Different combinations of variables.

Combining Like Terms: The Process

Combining like terms involves simplifying an algebraic expression by adding or subtracting the coefficients of like terms. The coefficient is the numerical factor of a term. For example, in the term 3x, the coefficient is 3.

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

1. Identify Like Terms: In this expression, we have two sets of like terms:

   • 3x and 2x
   • 3y and -7y

2. Combine the Coefficients: Add or subtract the coefficients of the like terms:

   • For the 'x' terms: 3 + 2 = 5
   • For the 'y' terms: 3 + (-7) = -4

3. Rewrite the Expression: Substitute the combined coefficients back into the expression:

   5x - 4y

Therefore, the simplified expression for 3x + 2x + 3y - 7y is 5x - 4y.

More Complex Examples

Let's explore more complex examples to demonstrate the versatility of combining like terms:

Example 1:

7a² + 3b - 2a² + 5b + 4

• Like terms: 7a², -2a²; 3b, 5b; 4 (constant)
• Combining: (7 - 2)a² + (3 + 5)b + 4
• Simplified Expression: 5a² + 8b + 4

Example 2:

-2xy + 5x - 3xy + 2x + y

• Like terms: -2xy, -3xy; 5x, 2x; y
• Combining: (-2 - 3)xy + (5 + 2)x + y
• Simplified Expression: -5xy + 7x + y

Example 3:

4x³ + 2x² - 3x + x³ + 5x² + 7x - 1

• Like terms: 4x³, x³; 2x², 5x²; -3x, 7x; -1 (constant)
• Combining: (4 + 1)x³ + (2 + 5)x² + (-3 + 7)x - 1
• Simplified Expression: 5x³ + 7x² + 4x - 1

Practical Applications

Combining like terms is a fundamental skill used extensively throughout algebra and beyond. Here are some practical applications:

• Solving Equations: Simplifying equations through combining like terms makes them easier to solve.
• Graphing: Simplifying equations simplifies the process of creating accurate graphs.
• Real-world problem-solving: Many real-world problems can be modeled using algebraic expressions. Simplifying these expressions makes it easier to find solutions. For example, calculating the total cost of items with varying quantities and prices often involves combining like terms.

Common Mistakes to Avoid

• Combining Unlike Terms: Remember, you can only combine terms with the exact same variables and exponents.
• Incorrect Sign Handling: Pay close attention to the signs (+ or -) before each term when adding or subtracting.
• Miscalculating Coefficients: Double-check your addition and subtraction of coefficients.

Practice Problems

To reinforce your understanding, try simplifying the following expressions by combining like terms:

1. 8a + 3b - 5a + 7b
2. 6x² - 2x + 4x² + 5x - 3
3. -4y³ + 2y² + 9y³ - 3y² + y
4. 2ab + 5a - 3ab + 2a - b
5. 5m²n + 3mn² - 2m²n + 4mn²

Answers: (Check your work after attempting the problems)

1. 3a + 10b
2. 10x² + 3x - 3
3. 5y³ - y² + y
4. -ab + 7a - b
5. 3m²n + 7mn²

Conclusion

Combining like terms is a cornerstone of algebraic manipulation. Mastering this skill is crucial for success in algebra and related fields. By understanding the principles, practicing consistently, and avoiding common mistakes, you can confidently simplify algebraic expressions and solve a wide range of mathematical problems. Remember to always meticulously identify like terms and accurately combine their coefficients. With consistent practice, you'll find combining like terms becomes second nature.