Check All Of The Solutions To The Original Equation

Checking All Solutions to the Original Equation: A Comprehensive Guide

Solving an equation is only half the battle. The real test lies in verifying whether your solutions are truly valid. This process, often overlooked, is crucial for ensuring accuracy and avoiding errors that can lead to incorrect conclusions. This article will delve into the intricacies of checking solutions, exploring various techniques and common pitfalls to help you master this essential mathematical skill. We'll cover linear equations, quadratic equations, radical equations, and even delve into the nuances of checking solutions in trigonometric and logarithmic equations.

Understanding the Importance of Solution Verification

Why bother checking solutions? Isn't finding them enough? Absolutely not. Several reasons highlight the critical importance of verifying your solutions:

• Identifying Errors: The process of checking allows you to catch mistakes made during the solution process. These mistakes can range from simple arithmetic errors to more complex algebraic mishaps. Verification acts as a powerful safeguard against such errors.

• Ensuring Accuracy: Without checking, there's always a lingering doubt about the accuracy of your answers. Verification provides the confidence that your solutions are indeed correct and satisfy the original equation.

• Avoiding Extraneous Solutions: Some mathematical operations, such as squaring both sides of an equation, can introduce extraneous solutions – solutions that appear valid during the solving process but don't actually satisfy the original equation. Checking solutions is the only way to identify and discard these extraneous solutions.

• Developing Mathematical Rigor: The habit of checking solutions instills a sense of mathematical rigor and precision. It fosters a more critical and analytical approach to problem-solving, leading to improved accuracy and a deeper understanding of the subject matter.

Checking Solutions: A Step-by-Step Approach

Regardless of the type of equation, the fundamental principle remains the same: substitute the obtained solutions back into the original equation and verify whether they satisfy the equation. Let's break this down step-by-step:

1. Isolate the variable: Before substitution, it's often helpful to isolate the variable you're solving for to simplify the process. This makes comparing the left and right sides of the equation easier.

2. Substitute the solution(s): Substitute each solution into the original equation, ensuring you replace all instances of the variable with the solution value.

3. Simplify both sides: Carefully simplify both sides of the equation independently. Use the order of operations (PEMDAS/BODMAS) to ensure accurate simplification.

4. Compare the results: Compare the simplified expressions on both sides of the equation. If the left-hand side equals the right-hand side, the solution is valid. If not, the solution is extraneous.

Examples: Checking Solutions in Different Equation Types

Let's illustrate this process with examples across various types of equations:

Linear Equations

Example: Solve and check the solution for the equation 3x + 5 = 14.

Solution:

1. Subtract 5 from both sides: 3x = 9
2. Divide both sides by 3: x = 3

Check:

Substitute x = 3 back into the original equation:

3(3) + 5 = 14 9 + 5 = 14 14 = 14

The solution x = 3 is valid.

Quadratic Equations

Example: Solve and check the solutions for the equation x² - 5x + 6 = 0.

Solution:

This equation can be factored as (x - 2)(x - 3) = 0. Therefore, the solutions are x = 2 and x = 3.

Check:

For x = 2:

(2)² - 5(2) + 6 = 0 4 - 10 + 6 = 0 0 = 0

For x = 3:

(3)² - 5(3) + 6 = 0 9 - 15 + 6 = 0 0 = 0

Both solutions, x = 2 and x = 3, are valid.

Radical Equations

Example: Solve and check the solution for the equation √(x + 2) = 3.

Solution:

1. Square both sides: x + 2 = 9
2. Subtract 2 from both sides: x = 7

Check:

Substitute x = 7 back into the original equation:

√(7 + 2) = 3 √9 = 3 3 = 3

The solution x = 7 is valid.

Equations with Absolute Values

Example: Solve and check the solutions for the equation |2x - 1| = 5.

Solution:

This equation has two cases:

• Case 1: 2x - 1 = 5 => 2x = 6 => x = 3
• Case 2: 2x - 1 = -5 => 2x = -4 => x = -2

Check:

For x = 3:

|2(3) - 1| = 5 |5| = 5 5 = 5

For x = -2:

|2(-2) - 1| = 5 |-5| = 5 5 = 5

Both solutions, x = 3 and x = -2, are valid.

Trigonometric Equations

Example: Solve and check the solutions for the equation sin(x) = 1/2 in the interval [0, 2π).

Solution:

The general solution is x = π/6 + 2kπ and x = 5π/6 + 2kπ, where k is an integer. In the interval [0, 2π), the solutions are x = π/6 and x = 5π/6.

Check:

For x = π/6:

sin(π/6) = 1/2 (True)

For x = 5π/6:

sin(5π/6) = 1/2 (True)

Both solutions are valid within the specified interval.

Logarithmic Equations

Example: Solve and check the solution for the equation log₂(x) = 3.

Solution:

By definition of logarithms, this equation is equivalent to 2³ = x, so x = 8.

Check:

Substitute x = 8 back into the original equation:

log₂(8) = 3 3 = 3

The solution x = 8 is valid.

Common Pitfalls to Avoid

• Arithmetic errors: Carefully perform all calculations to avoid simple mistakes.
• Algebraic errors: Double-check each step in your algebraic manipulations.
• Forgetting extraneous solutions: Always check your solutions, especially when dealing with radical equations or equations involving absolute values.
• Incorrect domain considerations: Remember to consider the domain of the equation, especially for logarithmic and trigonometric equations. Solutions outside the domain are invalid.
• Rounding errors: If you have rounded numbers during your calculation process, be mindful that this might affect the accuracy of your final result during the verification process. Ensure you maintain sufficient precision throughout the process to avoid substantial rounding errors affecting the final verification.

Conclusion

Checking solutions is not an optional step; it's an integral part of the problem-solving process. By diligently verifying your solutions, you ensure accuracy, identify errors, and build a stronger foundation in mathematics. The techniques and examples provided in this article offer a comprehensive guide to checking solutions in various types of equations. Mastering this skill will significantly enhance your mathematical abilities and confidence. Remember, a correct solution is only validated after it successfully passes the test of verification in the original equation. Always take the time to check your work – it's the hallmark of a true mathematician.