Ray's Mistake In Solving Equations A Step-by-Step Correction

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Ray attempted to solve the equation 3(x+6) = x+8+x, but made an error in his steps. Let's break down his solution, identify the mistake, and provide the correct solution. Understanding where errors occur in algebraic manipulations is crucial for mastering equation solving. This article aims to help you understand the common pitfalls and develop a systematic approach to solving equations accurately.

The Incorrect Solution

Ray's steps are as follows:

Equation:

3(x+6) = x+8+x

Step 1:

3x+18 = 2x+8

Step 2:

5x+18 = 8

Step 3:

5x = -10

Step 4:

x = -2

Identifying the Error

To pinpoint Ray's mistake, we need to meticulously examine each step. The initial step, Step 1, appears to be correct as the distributive property is applied properly: 3 * x + 3 * 6 = 3x + 18, and the right side of the equation, x + 8 + x, simplifies to 2x + 8. Therefore, 3x + 18 = 2x + 8 is a valid transformation.

The error lies in Step 2. To move from Step 1 (3x + 18 = 2x + 8) to the next step, Ray seems to have incorrectly added 2x to 3x, resulting in 5x. However, to isolate the 'x' terms on one side, the correct operation should have been subtracting 2x from both sides of the equation. This is a crucial step in maintaining the equation's balance and solving for 'x' accurately. By adding instead of subtracting, Ray fundamentally altered the equation, leading to an incorrect solution.

Detailed Error Explanation

The transition from Step 1 to Step 2 should have involved subtracting 2x from both sides of the equation to group the 'x' terms together. Ray's incorrect addition of 2x resulted in an unbalanced equation and a flawed path toward the solution. This highlights the importance of applying inverse operations correctly when solving algebraic equations.

Understanding why this error occurred is crucial for preventing it in the future. Misapplying algebraic operations, such as incorrectly adding terms instead of subtracting them, is a common mistake that can derail the entire solution process. It's essential to reinforce the fundamental principles of equation manipulation, particularly the concept of maintaining balance by performing the same operation on both sides.

The Correct Solution: A Step-by-Step Approach

To correctly solve the equation 3(x+6) = x+8+x, we need to follow a systematic approach, ensuring each step maintains the equation's balance. Let's go through the correct solution step by step:

Equation:

3(x+6) = x+8+x

Step 1: Apply the distributive property and combine like terms.

First, distribute the 3 on the left side:

3 * x + 3 * 6 = x + 8 + x

3x + 18 = x + 8 + x

Next, combine the 'x' terms on the right side:

3x + 18 = 2x + 8

This step correctly simplifies the equation while adhering to the order of operations and the distributive property.

Step 2: Isolate the 'x' terms on one side of the equation.

To do this, subtract 2x from both sides:

3x + 18 - 2x = 2x + 8 - 2x

x + 18 = 8

This step correctly isolates the 'x' term by using the inverse operation of addition (subtraction).

Step 3: Isolate the 'x' term by subtracting the constant.

Subtract 18 from both sides:

x + 18 - 18 = 8 - 18

x = -10

This step isolates 'x' completely, giving us the solution.

Step 4: Verification (Optional but recommended).

To ensure the solution is correct, substitute x = -10 back into the original equation:

3((-10) + 6) = (-10) + 8 + (-10)

3(-4) = -2 - 10

-12 = -12

Since the left side equals the right side, the solution x = -10 is correct.

Key Principles Used in the Correct Solution

  1. Distributive Property: Correctly distributing the 3 across the parentheses in the initial equation is crucial.
  2. Combining Like Terms: Simplifying both sides of the equation by combining 'x' terms and constants makes the equation easier to manipulate.
  3. Inverse Operations: Using inverse operations (subtraction for addition, division for multiplication) to isolate the variable is a fundamental principle of equation solving.
  4. Maintaining Balance: Performing the same operation on both sides of the equation ensures the equation remains balanced and the solution remains valid.
  5. Verification: Substituting the solution back into the original equation verifies its correctness, catching any potential errors made during the solving process.

Common Mistakes in Solving Equations

Understanding common mistakes helps in avoiding them. Here are some frequent errors made while solving algebraic equations:

  1. Incorrectly Applying the Distributive Property: Failing to distribute multiplication over addition or subtraction properly.
  2. Combining Unlike Terms: Attempting to add or subtract terms that are not like terms (e.g., 3x and 18).
  3. Incorrect Use of Inverse Operations: Applying the wrong operation to isolate the variable (e.g., adding when you should subtract).
  4. Not Maintaining Balance: Performing an operation on one side of the equation without doing the same on the other side.
  5. Sign Errors: Making mistakes with positive and negative signs, especially when distributing or combining terms.
  6. Forgetting the Order of Operations: Not following the correct order of operations (PEMDAS/BODMAS).
  7. Rushing through Steps: Skipping steps or solving mentally without writing it down can lead to mistakes.

Strategies to Avoid Mistakes

  • Write Each Step Clearly: Show all steps in your solution to minimize errors and make it easier to review your work.
  • Double-Check Your Work: Review each step to ensure you haven't made any mistakes, especially with signs and operations.
  • Use Parentheses: When substituting values or distributing, use parentheses to avoid sign errors.
  • Simplify Before Solving: Simplify each side of the equation as much as possible before attempting to isolate the variable.
  • Verify Your Solution: Substitute your solution back into the original equation to check for correctness.

Conclusion

Ray's error in Step 2 highlights the critical importance of correctly applying inverse operations when solving equations. By subtracting 2x from both sides instead of incorrectly adding, we arrived at the correct solution, x = -10. This step-by-step analysis emphasizes the need for meticulous attention to detail and a solid understanding of algebraic principles. Solving equations accurately requires a systematic approach, careful execution of each step, and a willingness to verify the solution. By understanding the common pitfalls and employing effective strategies, one can master the art of equation solving and confidently tackle more complex problems. Mastering equation solving is a foundational skill in mathematics, essential for success in algebra and beyond. Remember to always double-check your work, and practice consistently to build your proficiency. By identifying mistakes and understanding the correct methodologies, you enhance your problem-solving abilities and develop a deeper understanding of mathematical concepts.