Combining Like Terms Example Problems

Mastering the Art of Combining Like Terms: Examples and Explanations

Combining like terms is a fundamental algebraic skill crucial for simplifying expressions and solving equations. Understanding this concept unlocks the door to more advanced mathematical concepts. This comprehensive guide will walk you through the process, providing numerous examples and explanations to solidify your understanding. We'll explore various scenarios, from simple expressions to more complex ones, ensuring you gain confidence in tackling any problem involving like terms.

What are Like Terms?

Before diving into combining like terms, we need to understand what constitutes "like terms." 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 variable part must be identical.

Let's look at some examples:

Like Terms: 3x and 5x (both have the variable 'x' raised to the power of 1)
Like Terms: -2y² and 7y² (both have the variable 'y' raised to the power of 2)
Like Terms: 4ab and -9ab (both have the variables 'a' and 'b' raised to the power of 1)
Unlike Terms: 2x and 2y (different variables)
Unlike Terms: 3x² and 3x (different powers of the variable x)
Unlike Terms: 5a and 5ab (different variables)
Like Terms: 6, -2, and 10 (these are constants – they are considered like terms because they are all just numbers)

The Simple Rule: Combining Like Terms

The fundamental rule for combining like terms is remarkably straightforward: add or subtract the coefficients of the like terms, while keeping the variable part unchanged.

Let's illustrate this with some simple examples:

Example 1: Simplify 3x + 5x

Both terms are like terms (they both have 'x').
Add the coefficients: 3 + 5 = 8
Keep the variable part: x
Simplified expression: 8x

Example 2: Simplify 7y² - 2y²

Both terms are like terms (they both have 'y²').
Subtract the coefficients: 7 - 2 = 5
Keep the variable part: y²
Simplified expression: 5y²

Example 3: Simplify 4ab + 6ab - 2ab

All terms are like terms (they all have 'ab').
Add and subtract the coefficients: 4 + 6 - 2 = 8
Keep the variable part: ab
Simplified expression: 8ab

Combining Like Terms with Multiple Variables and Exponents

Now, let's tackle examples with more complexity, involving multiple variables and exponents:

Example 4: Simplify 2x²y + 5x²y - 3x²y

All terms are like terms (they all have 'x²y').
Add and subtract the coefficients: 2 + 5 - 3 = 4
Keep the variable part: x²y
Simplified expression: 4x²y

Example 5: Simplify 3a²b - 2ab² + 5a²b + ab²

We have two sets of like terms here: 3a²b and 5a²b; and -2ab² and ab².
Combine the a²b terms: 3a²b + 5a²b = 8a²b
Combine the ab² terms: -2ab² + ab² = -ab²
The simplified expression is 8a²b - ab²

Example 6: Simplify 4x³ - 2x² + 7x³ + 5x² - 3x

Group like terms: (4x³ + 7x³) + (-2x² + 5x²) - 3x
Combine x³ terms: 4x³ + 7x³ = 11x³
Combine x² terms: -2x² + 5x² = 3x²
The -3x term remains unchanged.
Simplified expression: 11x³ + 3x² - 3x

Combining Like Terms with Parentheses

Parentheses can add another layer of complexity. Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction.

Example 7: Simplify 2(3x + 4y) + 5x - 2y

First, distribute the 2 to the terms inside the parentheses: 6x + 8y + 5x - 2y
Now, combine like terms: (6x + 5x) + (8y - 2y)
Simplified expression: 11x + 6y

Example 8: Simplify 3(x² - 2x + 1) - 2(x² + x - 3)

Distribute the 3 and -2: 3x² - 6x + 3 - 2x² - 2x + 6
Combine like terms: (3x² - 2x²) + (-6x - 2x) + (3 + 6)
Simplified expression: x² - 8x + 9

Combining Like Terms with Fractions

Fractions don't change the core principle; you still add or subtract the coefficients.

Example 9: Simplify (1/2)x + (3/4)x

Find a common denominator for the coefficients: (2/4)x + (3/4)x
Add the coefficients: (2/4 + 3/4)x = (5/4)x
Simplified expression: (5/4)x or 1.25x

Example 10: Simplify (2/3)y² - (1/6)y² + y

Find a common denominator for the y² terms: (4/6)y² - (1/6)y² + y
Combine the y² terms: (3/6)y² + y
Simplified expression: (1/2)y² + y

Word Problems Involving Combining Like Terms

Let's see how combining like terms applies in real-world scenarios:

Example 11: John buys 3 apples and 2 oranges. Later, he buys 5 more apples and 1 orange. How many apples and oranges does John have in total?

Apples: 3 + 5 = 8 apples
Oranges: 2 + 1 = 3 oranges
Total: 8 apples and 3 oranges This is a simple example of combining like terms without explicit algebraic notation, showing its underlying application.

Example 12: A rectangle has a length of (2x + 3) units and a width of (x - 1) units. Find the perimeter of the rectangle.

Perimeter = 2(length) + 2(width)
Perimeter = 2(2x + 3) + 2(x - 1)
Distribute: 4x + 6 + 2x - 2
Combine like terms: (4x + 2x) + (6 - 2)
Perimeter = 6x + 4 units

Frequently Asked Questions (FAQ)

Q1: What happens if I have unlike terms?

A1: You cannot combine unlike terms. They must remain separate in the simplified expression.

Q2: Can I combine like terms in any order?

A2: Yes, the commutative property of addition allows you to rearrange terms before combining like terms.

Q3: What if I have a negative coefficient?

A3: Treat negative coefficients just like you would any other number. Remember the rules for adding and subtracting integers.

Q4: Is there a software or online tool to help with combining like terms?

A4: While many online calculators can simplify expressions, understanding the process manually is crucial for developing your algebraic skills.

Conclusion

Combining like terms is a fundamental building block in algebra. Mastering this skill is essential for simplifying expressions, solving equations, and tackling more advanced mathematical problems. By practicing the examples provided and understanding the underlying principles, you will build a strong foundation for future success in algebra and beyond. Remember to focus on identifying like terms, paying close attention to variables and their exponents. Consistent practice and attention to detail are key to becoming proficient in combining like terms.