Error In Equation Solving Identifying The Mistake In A Step-by-Step Solution

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In the realm of mathematics, solving equations is a fundamental skill. However, even the most seasoned mathematicians can sometimes make mistakes. In this article, we will dissect a step-by-step solution to an equation, pinpointing the exact location where an error occurred. This exercise not only sharpens our problem-solving abilities but also reinforces the importance of meticulousness in mathematical calculations. Let's embark on this journey of mathematical scrutiny!

The Problematic Equation and Its Solution Attempt

The Initial Equation

Our starting point is the equation:

3(x + 2) - 2x = 9

This equation involves variables, constants, and operations, making it a typical algebraic expression. The goal is to isolate the variable 'x' and determine its value.

The Step-by-Step Solution

Now, let's examine the attempted solution, line by line, to identify any missteps:

Step 1:

3x + 6 - 2x = 9

This step involves distributing the '3' across the parentheses. It seems correct as 3 * x = 3x and 3 * 2 = 6. So far, so good.

Step 2:

+2x + 2x

Here's where things get interesting. The intention seems to be isolating the 'x' terms, but this step is presented in isolation. It's unclear what operation is being performed and on which side of the equation. This is a potential area of concern.

Step 3:

5x + 6 = 9

This step appears to be a combination of the previous steps. Let's analyze how this result was obtained. If we combine the '3x' and '-2x' terms from Step 1, we get 'x'. However, the equation shows '5x', which indicates a possible error in combining like terms or an incorrect operation performed in the previous step. This is a critical point to investigate further.

Step 4:

-6 -6

Similar to Step 2, this step is presented in isolation. It suggests an attempt to subtract '6' from both sides of the equation, which is a valid algebraic manipulation. However, without the context of the full equation, it's hard to judge its correctness.

Step 5:

... (Incomplete)

The solution is incomplete, but we have enough information to pinpoint the error.

Identifying the Mistake A Deep Dive

The Critical Error A Detailed Explanation

The error likely occurs between Step 2 and Step 3. Step 2's isolated +2x + 2x suggests an attempt to add 2x to both sides, but this operation doesn't logically follow from Step 1. The correct approach after Step 1 would be to combine the 'x' terms (3x and -2x) directly. Combining like terms in the initial equation 3x + 6 - 2x = 9 should result in x + 6 = 9, not 5x + 6 = 9. The jump to 5x indicates an arithmetic error or a misunderstanding of how to combine like terms.

Why This Mistake Matters

This seemingly small error has significant consequences. It leads to an incorrect equation, which will ultimately result in a wrong solution for 'x'. This highlights the importance of carefully reviewing each step in the equation-solving process and ensuring that each operation is mathematically sound.

The Correct Steps to Solve the Equation

To solve the equation correctly, we should follow these steps:

  1. Distribute: Expand the parentheses: 3(x + 2) - 2x = 9 becomes 3x + 6 - 2x = 9.
  2. Combine Like Terms: Combine the 'x' terms: 3x - 2x + 6 = 9 simplifies to x + 6 = 9.
  3. Isolate the Variable: Subtract '6' from both sides: x + 6 - 6 = 9 - 6, which gives x = 3.

Therefore, the correct solution for 'x' is 3.

The Importance of Careful Execution

Avoiding Common Pitfalls

This example illustrates a common pitfall in algebra: making errors when combining like terms or performing algebraic manipulations. To avoid such errors, it's crucial to:

  • Double-check each step: Before moving on, ensure that the previous operation was performed correctly.
  • Write clearly: A neat and organized presentation can help prevent errors.
  • Understand the underlying principles: A solid grasp of algebraic principles is essential for accurate equation solving.
  • Practice regularly: Consistent practice builds confidence and reduces the likelihood of mistakes.

The Role of Meticulousness in Mathematics

Meticulousness is a cornerstone of mathematical accuracy. Every symbol, every operation, and every step must be executed with precision. This example serves as a reminder that even small errors can propagate through the solution, leading to incorrect results. By developing a habit of carefulness and attention to detail, we can minimize the risk of making such mistakes.

Conclusion Mastering Equation Solving

Key Takeaways

In conclusion, the mistake in the provided solution occurred in the transition between Step 2 and Step 3, where the 'x' terms were incorrectly combined. This error highlights the importance of carefully combining like terms and double-checking each step in the equation-solving process. By understanding the principles of algebra and practicing diligently, we can enhance our equation-solving skills and minimize the risk of errors.

Final Thoughts

Solving equations is not just about finding the right answer; it's about developing a logical and methodical approach to problem-solving. By learning from mistakes and reinforcing correct techniques, we can become more confident and proficient mathematicians. Remember, every error is an opportunity to learn and grow. Embrace the challenge, stay meticulous, and enjoy the journey of mathematical discovery! This analytical approach not only helps in identifying errors but also reinforces the core principles of equation-solving, making it a valuable exercise for students and math enthusiasts alike.

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Identify the step in the given solution where a mistake occurred while solving the equation 3(x+2) - 2x = 9.

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Error in Equation Solving Identifying the Mistake in a Step-by-Step Solution