Lesson 3.4 Solving Complex 1-variable Equations Answer Key

Lesson 3.4: Solving Complex 1-Variable Equations – A Comprehensive Guide

This guide delves into the intricacies of solving complex one-variable equations, providing a step-by-step approach, numerous examples, and strategies to tackle various equation types. While we won't provide a specific "answer key" tied to a particular textbook (as those are copyright protected), this comprehensive guide will equip you with the skills to solve any complex one-variable equation you encounter. We'll cover techniques such as distributing, combining like terms, and working with fractions and decimals effectively.

Understanding the Fundamentals

Before tackling complex equations, let's solidify our understanding of the basics. A one-variable equation contains only one unknown variable (usually represented by x, y, or another letter). The goal is to isolate this variable on one side of the equation to find its value. This involves applying inverse operations to maintain the equation's balance.

Key Principles:

• The Addition Property of Equality: Adding the same number to both sides of an equation maintains equality. If a = b, then a + c = b + c.
• The Subtraction Property of Equality: Subtracting the same number from both sides of an equation maintains equality. If a = b, then a – c = b – c.
• The Multiplication Property of Equality: Multiplying both sides of an equation by the same non-zero number maintains equality. If a = b, then ac = bc (where c ≠ 0).
• The Division Property of Equality: Dividing both sides of an equation by the same non-zero number maintains equality. If a = b, then a/c = b/c (where c ≠ 0).

Tackling Complex Equations: A Step-by-Step Approach

Complex one-variable equations involve multiple steps and may include parentheses, fractions, decimals, and like terms. Here's a systematic approach:

Step 1: Simplify Both Sides of the Equation

This involves:

• Distributing: If parentheses are present, distribute any number or variable outside the parentheses to each term inside. For example: 3(x + 2) becomes 3x + 6.
• Combining Like Terms: Combine terms with the same variable raised to the same power. For instance, 2x + 5x simplifies to 7x. Similarly, constant terms (numbers without variables) are combined.

Example: Solve 2(x + 3) + 4x = 18

1. Distribute: 2x + 6 + 4x = 18
2. Combine Like Terms: 6x + 6 = 18

Step 2: Isolate the Variable Term

This step focuses on moving all terms containing the variable to one side of the equation and all constant terms to the other side. Use the addition and subtraction properties of equality to achieve this.

Continuing the Example:

3. Subtract 6 from both sides: 6x + 6 - 6 = 18 - 6 => 6x = 12

Step 3: Solve for the Variable

Use the multiplication or division property of equality to isolate the variable completely.

Continuing the Example:

4. Divide both sides by 6: 6x/6 = 12/6 => x = 2

Step 4: Verify Your Solution

Substitute your solution back into the original equation to check if it's correct.

Verifying the Example:

2(2 + 3) + 4(2) = 2(5) + 8 = 10 + 8 = 18. The solution x = 2 is correct.

Dealing with Fractions and Decimals

Equations involving fractions and decimals require additional steps:

Working with Fractions:

1. Find the Least Common Denominator (LCD): Find the smallest number that all denominators divide into evenly.
2. Multiply all terms by the LCD: This eliminates the fractions.
3. Follow Steps 1-4 (above) to solve the resulting equation.

Example: Solve (1/2)x + (1/3) = 5/6

1. Find LCD: The LCD of 2, 3, and 6 is 6.
2. Multiply by LCD: 6 * [(1/2)x + (1/3)] = 6 * (5/6) => 3x + 2 = 5
3. Solve: 3x = 3 => x = 1

Working with Decimals:

1. Multiply all terms by a power of 10 to eliminate the decimals: For example, multiply by 10 to eliminate one decimal place, 100 for two decimal places, and so on.
2. Follow Steps 1-4 (above) to solve the resulting equation.

Example: Solve 0.2x + 0.5 = 1.7

1. Multiply by 10: 10(0.2x + 0.5) = 10(1.7) => 2x + 5 = 17
2. Solve: 2x = 12 => x = 6

Advanced Techniques and Equation Types

Let's explore some more challenging equation types:

Equations with Variables on Both Sides:

Collect variable terms on one side and constant terms on the other.

Example: Solve 5x + 7 = 2x + 16

1. Subtract 2x from both sides: 3x + 7 = 16
2. Subtract 7 from both sides: 3x = 9
3. Divide by 3: x = 3

Equations with Absolute Value:

Remember that the absolute value of a number is its distance from zero, always non-negative. Solve for both the positive and negative cases.

Example: Solve |x - 3| = 5

1. Positive case: x - 3 = 5 => x = 8
2. Negative case: x - 3 = -5 => x = -2

Equations with Square Roots:

Isolate the square root and then square both sides to eliminate the radical. Remember to check for extraneous solutions (solutions that don't satisfy the original equation).

Example: Solve √(x + 2) = 4

1. Square both sides: x + 2 = 16
2. Solve: x = 14 (Check: √(14+2) = √16 = 4. The solution is valid.)

Quadratic Equations:

These involve the variable raised to the power of 2 (x²). Solving techniques include factoring, the quadratic formula, or completing the square.

Example (Factoring): Solve x² + 5x + 6 = 0

(x + 2)(x + 3) = 0 => x = -2 or x = -3

Practice and Mastery

Solving complex one-variable equations requires practice. Work through numerous examples, focusing on each step systematically. Start with simpler equations and gradually increase the complexity. Don't hesitate to review the fundamental principles and techniques as needed. The more you practice, the more confident and proficient you'll become. Remember to always check your solutions by substituting them back into the original equation. This step is crucial to ensure accuracy and identify any errors made during the solving process. By consistently applying these strategies and dedicating time to practice, you will master the art of solving complex one-variable equations and gain a deeper understanding of algebraic concepts.