Combining Like Terms And Solving Equations

Mastering Algebra: Combining Like Terms and Solving Equations

This comprehensive guide delves into the fundamental algebraic concepts of combining like terms and solving equations. Understanding these two interconnected skills is crucial for success in higher-level mathematics and various STEM fields. We'll explore these concepts in detail, providing numerous examples and practical applications to solidify your understanding. This article will cover everything from the basics to more complex scenarios, ensuring you gain a solid grasp of these essential algebraic tools.

What are Like Terms?

Before tackling the process of combining like terms, it's crucial to understand what constitutes "like terms." Like terms are terms that have the same variables raised to the same powers. The coefficients (the numbers multiplying the variables) can be different, but the variables and their exponents must be identical.

Here are some examples to illustrate:

• Like terms: 3x and 7x (same variable, same exponent)
• Like terms: -5y² and 2y² (same variable, same exponent)
• Like terms: 1/2ab and -3ab (same variables, same exponents)
• Unlike terms: 2x and 2y (different variables)
• Unlike terms: 4x² and 4x (same variable, different exponents)
• Unlike terms: 6x and 6 (one is a constant, the other has a variable)

Understanding the distinction between like and unlike terms is fundamental to simplifying algebraic expressions.

Combining Like Terms: A Step-by-Step Guide

Combining like terms is the process of simplifying an algebraic expression by adding or subtracting terms that are alike. The process involves identifying like terms, adding or subtracting their coefficients, and retaining the common variable and exponent.

Let's illustrate with a few examples:

Example 1: Simplify 5x + 2x - 3x

• Identify like terms: All terms are like terms (they all contain 'x').
• Combine coefficients: 5 + 2 - 3 = 4
• Result: 4x

Example 2: Simplify 7y² - 4y² + y

• Identify like terms: 7y² and -4y² are like terms. 'y' is a separate term.
• Combine coefficients of like terms: 7 - 4 = 3
• Result: 3y² + y (Note: we cannot combine 3y² and y further as they are unlike terms)

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

• Identify like terms: 2a²b and -a²b are like terms. 5ab² and 3ab² are like terms.
• Combine coefficients of like terms: 2 - 1 = 1; 5 + 3 = 8
• Result: a²b + 8ab²

Example 4: Combining Like Terms with Parentheses

Often, you'll encounter expressions with parentheses. Remember to distribute any coefficients before combining like terms.

Simplify: 3(2x + y) - 2(x - 3y)

1. Distribute: 6x + 3y - 2x + 6y
2. Combine like terms: (6x - 2x) + (3y + 6y)
3. Simplify: 4x + 9y

Solving Linear Equations: The Basics

Solving an equation means finding the value of the variable that makes the equation true. Linear equations are equations where the highest power of the variable is 1. The goal is to isolate the variable on one side of the equation. This involves using inverse operations (addition/subtraction, multiplication/division) to maintain balance and simplify.

Fundamental Principles:

• Addition Property of Equality: Adding the same number to both sides of an equation maintains equality.
• Subtraction Property of Equality: Subtracting the same number from both sides of an equation maintains equality.
• Multiplication Property of Equality: Multiplying both sides of an equation by the same non-zero number maintains equality.
• Division Property of Equality: Dividing both sides of an equation by the same non-zero number maintains equality.

Example 5: Solve for x: x + 5 = 12

• Subtract 5 from both sides: x + 5 - 5 = 12 - 5
• Simplify: x = 7

Example 6: Solve for y: y - 3 = 8

• Add 3 to both sides: y - 3 + 3 = 8 + 3
• Simplify: y = 11

Example 7: Solve for z: 3z = 15

• Divide both sides by 3: 3z / 3 = 15 / 3
• Simplify: z = 5

Example 8: Solve for w: w / 4 = 6

• Multiply both sides by 4: (w / 4) * 4 = 6 * 4
• Simplify: w = 24

Solving Equations with Multiple Steps and Like Terms

Many equations require a combination of steps. Here's where combining like terms plays a vital role.

Example 9: Solve for x: 2x + 5 = x + 11

1. Subtract x from both sides: 2x - x + 5 = x - x + 11
2. Simplify: x + 5 = 11
3. Subtract 5 from both sides: x + 5 - 5 = 11 - 5
4. Simplify: x = 6

Example 10: Solve for y: 3y - 7 = 2y + 4

1. Subtract 2y from both sides: 3y - 2y - 7 = 2y - 2y + 4
2. Simplify: y - 7 = 4
3. Add 7 to both sides: y - 7 + 7 = 4 + 7
4. Simplify: y = 11

Example 11: Solving Equations with Parentheses and Like Terms

Solve for x: 2(x + 3) = 4x - 2

1. Distribute: 2x + 6 = 4x - 2
2. Subtract 2x from both sides: 2x - 2x + 6 = 4x - 2x - 2
3. Simplify: 6 = 2x - 2
4. Add 2 to both sides: 6 + 2 = 2x - 2 + 2
5. Simplify: 8 = 2x
6. Divide both sides by 2: 8 / 2 = 2x / 2
7. Simplify: x = 4

Example 12: Solving Equations with Fractions

Solve for y: (1/2)y + 3 = 7

1. Subtract 3 from both sides: (1/2)y + 3 - 3 = 7 - 3
2. Simplify: (1/2)y = 4
3. Multiply both sides by 2: 2 * (1/2)y = 4 * 2
4. Simplify: y = 8

Solving Equations with Variables on Both Sides: Advanced Techniques

When variables appear on both sides of the equation, the process involves strategically moving terms to isolate the variable. The core principles remain the same: maintain balance by performing the same operation on both sides. Remember to always prioritize simplifying the equation by combining like terms before attempting to isolate the variable.

Example 13: Solve for x: 5x + 3 = 2x + 9

1. Subtract 2x from both sides: 5x - 2x + 3 = 2x - 2x + 9
2. Simplify: 3x + 3 = 9
3. Subtract 3 from both sides: 3x + 3 - 3 = 9 - 3
4. Simplify: 3x = 6
5. Divide both sides by 3: 3x / 3 = 6 / 3
6. Simplify: x = 2

Example 14: Solve for y: 4(y - 1) = 2(y + 3)

1. Distribute: 4y - 4 = 2y + 6
2. Subtract 2y from both sides: 4y - 2y - 4 = 2y - 2y + 6
3. Simplify: 2y - 4 = 6
4. Add 4 to both sides: 2y - 4 + 4 = 6 + 4
5. Simplify: 2y = 10
6. Divide both sides by 2: 2y / 2 = 10 / 2
7. Simplify: y = 5

Conclusion

Combining like terms and solving equations are fundamental algebraic skills with far-reaching applications. By mastering these techniques, you'll build a strong foundation for more advanced mathematical concepts. Remember to practice regularly, work through diverse examples, and break down complex problems into smaller, manageable steps. The key to success lies in understanding the underlying principles and consistently applying them. Through diligent practice, these essential algebraic tools will become second nature, enabling you to confidently approach and solve a wide range of mathematical problems.