Solving Equations By Combining Like Terms

Solving Equations by Combining Like Terms: A Comprehensive Guide

Meta Description: Mastering algebra requires understanding how to solve equations. This comprehensive guide explains solving equations by combining like terms, covering various examples and techniques for beginners and advanced learners. Learn to simplify equations and find solutions efficiently.

Solving equations is a fundamental skill in algebra. While seemingly simple at first glance, the ability to effectively solve equations forms the bedrock of more complex mathematical concepts. One of the most crucial techniques in this process is combining like terms. This method simplifies equations, making them easier to solve and leading to a more efficient and accurate solution. This guide provides a comprehensive overview of solving equations by combining like terms, catering to both beginners grappling with the basics and more advanced students seeking to refine their skills.

Understanding Like Terms

Before diving into solving equations, we need a clear understanding of what constitutes "like terms." Like terms are terms in an algebraic expression 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 parts must be identical.

Here are some examples:

• Like Terms: 3x and 7x (both have the variable 'x' raised to the power of 1)
• Like Terms: -5y² and 2y² (both have the variable 'y' raised to the power of 2)
• Like Terms: 4ab and -2ab (both have the variables 'a' and 'b' raised to the power of 1)
• Unlike Terms: 2x and 2y (different variables)
• Unlike Terms: 5x² and 5x (different powers of the variable 'x')
• Unlike Terms: 6a and 6a²b (different variables and powers)

Combining Like Terms: The Basics

Combining like terms involves adding or subtracting the coefficients of terms with the same variables and powers. The variable part remains unchanged.

Example 1: Simplify the expression 3x + 7x.

Since both terms have the variable 'x' raised to the power of 1, they are like terms. We simply add the coefficients: 3 + 7 = 10. Therefore, the simplified expression is 10x.

Example 2: Simplify the expression 5y² - 2y² + 4y².

All terms have the variable 'y' raised to the power of 2. We add and subtract the coefficients: 5 - 2 + 4 = 7. The simplified expression is 7y².

Example 3: Simplify the expression 2ab + 5ab - 3ab.

All terms have the variables 'a' and 'b' raised to the power of 1. We add and subtract the coefficients: 2 + 5 - 3 = 4. The simplified expression is 4ab.

Solving One-Step Equations by Combining Like Terms

Once you understand combining like terms, you can apply it to solve simple one-step equations. A one-step equation involves a single operation (addition, subtraction, multiplication, or division) to isolate the variable.

Example 4: Solve the equation 2x + 5x = 21.

First, combine the like terms on the left side: 2x + 5x = 7x. The equation becomes 7x = 21. To solve for x, divide both sides by 7: x = 3.

Example 5: Solve the equation 8y - 3y = 15.

Combine like terms: 8y - 3y = 5y. The equation becomes 5y = 15. Divide both sides by 5: y = 3.

Solving Two-Step Equations by Combining Like Terms

Two-step equations require two operations to isolate the variable. Combining like terms is often the first step in solving these equations.

Example 6: Solve the equation 3x + 5x + 7 = 25.

First, combine the like terms: 3x + 5x = 8x. The equation becomes 8x + 7 = 25. Subtract 7 from both sides: 8x = 18. Divide both sides by 8: x = 18/8 = 9/4 or 2.25.

Example 7: Solve the equation 6y - 2y - 4 = 12.

Combine like terms: 6y - 2y = 4y. The equation becomes 4y - 4 = 12. Add 4 to both sides: 4y = 16. Divide both sides by 4: y = 4.

Solving Equations with Like Terms on Both Sides

When like terms appear on both sides of the equation, you need to simplify each side separately before combining like terms across the equation.

Example 8: Solve the equation 2x + 5 = 3x + 1.

First, we can subtract 2x from both sides to get all x terms on one side: 5 = x + 1. Then subtract 1 from both sides: x = 4.

Example 9: Solve the equation 4y - 7 = 2y + 3.

Subtract 2y from both sides: 2y - 7 = 3. Add 7 to both sides: 2y = 10. Divide both sides by 2: y = 5.

Equations with Parentheses and Combining Like Terms

Equations often contain parentheses. Before combining like terms, you must distribute any terms outside the parentheses to the terms inside.

Example 10: Solve the equation 2(x + 3) + 4x = 18.

Distribute the 2 to both terms inside the parentheses: 2x + 6 + 4x = 18. Combine like terms: 6x + 6 = 18. Subtract 6 from both sides: 6x = 12. Divide both sides by 6: x = 2.

Example 11: Solve the equation 3(2y - 1) - y = 11.

Distribute the 3: 6y - 3 - y = 11. Combine like terms: 5y - 3 = 11. Add 3 to both sides: 5y = 14. Divide both sides by 5: y = 14/5 or 2.8.

Equations with Fractions and Combining Like Terms

Equations can also involve fractions. To simplify, find a common denominator and combine like terms.

Example 12: Solve the equation (1/2)x + (1/4)x = 6.

Find a common denominator (4): (2/4)x + (1/4)x = 6. Combine like terms: (3/4)x = 6. Multiply both sides by 4/3: x = 8.

Example 13: Solve the equation (2/3)y - (1/6)y = 5.

Find a common denominator (6): (4/6)y - (1/6)y = 5. Combine like terms: (3/6)y = 5, which simplifies to (1/2)y = 5. Multiply both sides by 2: y = 10.

Advanced Techniques and Problem Solving Strategies

While combining like terms is a fundamental technique, solving more complex equations often requires a combination of strategies. Here are some advanced considerations:

• Order of Operations (PEMDAS/BODMAS): Always follow the order of operations (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction) when simplifying expressions.
• Strategic Rearrangement: Sometimes, rearranging the equation to group like terms strategically can simplify the solving process.
• Factoring: In more advanced equations, factoring can be essential to simplify and solve.
• Using Properties of Equality: Remember that you can add, subtract, multiply, or divide both sides of an equation by the same number (excluding division by zero) without changing the solution.

Conclusion

Solving equations by combining like terms is a fundamental skill in algebra. By mastering this technique, you build a strong foundation for tackling more advanced algebraic concepts. Remember to focus on understanding like terms, applying the correct order of operations, and employing strategic simplification techniques to solve equations efficiently and accurately. Practice is key – the more you solve equations, the more comfortable and confident you'll become in your ability to manipulate algebraic expressions and find solutions. Consistent practice with diverse examples will solidify your understanding and improve your problem-solving skills, empowering you to tackle increasingly complex algebraic challenges. Remember to always check your solutions by substituting them back into the original equation to verify their accuracy.