Identifying Mistakes In Solving Linear Equations A Step-by-Step Analysis

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In the realm of mathematics, accuracy is paramount, especially when solving equations. A small error in one step can lead to a completely incorrect solution. This article delves into a specific linear equation problem, meticulously examining each step to pinpoint the exact mistake made. By understanding where errors commonly occur, we can strengthen our problem-solving skills and approach equations with greater confidence.

The Problem: Solving the Equation

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Let's consider the following equation and the steps taken to solve it:

6x - 1 = -2x + 9
8x - 1 = 9
8x = 10
x = 8/10
x = 4/5

The task is to identify the mistake within these steps. Each line represents a transformation of the equation, aiming to isolate the variable 'x'. We need to carefully analyze each step to ensure that the mathematical operations performed are valid and maintain the equality.

Step-by-Step Analysis

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To accurately find the mistake, let's break down each step and justify the operations performed:

  1. Original Equation: 6x - 1 = -2x + 9

    This is the starting point. No operations have been performed yet.

  2. Step 1: 8x - 1 = 9

    Here, it appears that the solver added 2x to both sides of the equation. This is a valid application of the addition property of equality. Adding the same quantity to both sides maintains the balance of the equation.

    • Original equation: 6x - 1 = -2x + 9
    • Adding 2x to both sides: 6x + 2x - 1 = -2x + 2x + 9
    • Simplified: 8x - 1 = 9 So far, the step is correct.

  3. Step 2: 8x = 10

    In this step, the solver seems to have added 1 to both sides of the equation. Again, this is a correct application of the addition property of equality.

    • Previous equation: 8x - 1 = 9
    • Adding 1 to both sides: 8x - 1 + 1 = 9 + 1
    • Simplified: 8x = 10 This step is also correct.

  4. Step 3: x = 8/10

    This is where the mistake occurs. To isolate 'x', we need to divide both sides of the equation by the coefficient of 'x', which is 8. However, the solver incorrectly placed 8 in the numerator and 10 in the denominator. The correct operation should be dividing both sides by 8.

    • Previous equation: 8x = 10
    • Dividing both sides by 8: (8x) / 8 = 10 / 8
    • Correct result: x = 10/8

  5. Step 4: x = 4/5

    This step simplifies the incorrect fraction 8/10. While the simplification is done correctly, it's based on the incorrect result from the previous step. Therefore, this answer is also incorrect.

The Correct Solution

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To reiterate, the mistake was made in Step 3, where the solver incorrectly divided to isolate 'x'. Let's solve the equation correctly:

  1. 6x - 1 = -2x + 9
  2. 8x - 1 = 9
  3. 8x = 10
  4. x = 10/8
  5. Simplify the fraction: x = 5/4

Therefore, the correct solution is x = 5/4, not x = 4/5.

Understanding the Addition Property of Equality

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The addition property of equality is a fundamental concept in algebra. It states that if you add the same value to both sides of an equation, the equation remains balanced. Mathematically, it can be expressed as:

If a = b, then a + c = b + c

Where a, b, and c are any mathematical expressions. This property is crucial for manipulating equations and isolating variables. In the given problem, the addition property of equality was correctly applied in the initial steps to move terms around and simplify the equation.

Common Mistakes and How to Avoid Them

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Several common mistakes can occur when solving equations. Recognizing these pitfalls can help prevent errors:

  • Incorrectly Applying the Order of Operations: Always follow the order of operations (PEMDAS/BODMAS) when simplifying expressions. This ensures consistency and accuracy.
  • Not Distributing Negatives Properly: When dealing with expressions involving parentheses and negative signs, ensure that the negative sign is distributed correctly to all terms inside the parentheses.
  • Combining Like Terms Incorrectly: Only combine terms that have the same variable and exponent. For example, 3x and 5x can be combined, but 3x and 5x² cannot.
  • Dividing or Multiplying by Zero: Division by zero is undefined. Avoid this operation at all costs. Multiplying both sides of an equation by zero will make the equation trivial (0 = 0) and lose information.
  • Making Arithmetic Errors: Simple arithmetic mistakes, such as incorrect addition or subtraction, can lead to wrong answers. Double-check your calculations, especially in multi-step problems.
  • Incorrectly Dividing to Isolate Variables: This was the exact mistake made in the problem we analyzed. Remember to divide both sides by the coefficient of the variable you want to isolate.

To avoid these mistakes, practice is essential. Work through various problems, showing each step clearly. If you make an error, try to understand why it happened and how to correct it. Review the fundamental properties and rules of algebra regularly.

The Importance of Showing Your Work

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In mathematics, showing your work is just as important as getting the correct answer. Writing out each step in the solution process provides several benefits:

  • Clarity and Organization: Showing your work helps you organize your thoughts and present your solution logically. It makes it easier to follow your reasoning.
  • Error Detection: By writing down each step, you can more easily identify mistakes you might have made. This is crucial for self-correction and learning.
  • Partial Credit: In an academic setting, even if your final answer is incorrect, showing your work can earn you partial credit if the steps you took were mathematically sound.
  • Communication: Showing your work allows others (teachers, classmates, colleagues) to understand your approach and provide feedback.

In the context of the problem we analyzed, writing out each step clearly revealed the error in Step 3. If the solver had only written the final answer, the mistake might have gone unnoticed.

Practice Problems

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To solidify your understanding of solving equations and avoiding common mistakes, try solving the following problems. Be sure to show your work!

  1. 4x + 3 = 2x - 7
  2. 3(y - 2) = 5y + 4
  3. (z/2) + 1 = (z/3) - 2

By working through these problems and carefully analyzing each step, you can build your problem-solving skills and gain confidence in algebra.

Conclusion

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Solving equations accurately requires a solid understanding of mathematical principles and careful attention to detail. The problem we analyzed highlights the importance of correctly applying operations and avoiding common mistakes, especially when dividing to isolate variables. The addition property of equality is a powerful tool, but it must be used in conjunction with other rules and strategies. By showing your work, practicing regularly, and learning from your errors, you can become a more proficient problem solver in mathematics and beyond.