Unveiling The Error: Solving Equations Step-by-Step

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Unveiling the Error: Solving Equations Step-by-Step

Hey math enthusiasts! Today, we're diving into the world of algebraic equations. Our goal? To pinpoint the step where things go south. We'll break down the problem, meticulously examine each stage, and hopefully, have a "aha!" moment where we catch the mistake. Let's get started, shall we?

Decoding the Equation: A Step-by-Step Breakdown

The problem at hand is: 2n - 9 = 3(n - 2). Our aim is to isolate 'n' and find its value. So, let's look at each step provided and understand what's happening. Each step represents a specific operation to simplify the equation and get us closer to the solution. The most important thing here is to understand the rules and order of operations. Remember that we need to keep the equation balanced by applying the same operation to both sides.

Step 1: Expanding the Expression

  • Original equation: 2n - 9 = 3(n - 2)
  • Step 1: 2n - 9 = 3n - 6

This step involves distributing the 3 across the terms inside the parentheses on the right side of the equation. We multiply 3 by 'n' and then 3 by -2. The result should be 3n - 6. Now, let's take a closer look. Is this step correct? Yes, it is. The distributive property has been correctly applied. No issues here, guys!

We successfully expanded the right side of the equation. This is the foundation upon which the rest of our steps will be built. So far, so good. Remember that careful attention to detail is critical here; a small miscalculation can lead to a completely incorrect answer. We should always check our work to avoid unnecessary errors. The first step involves distributing the 3 across the terms within the parentheses, resulting in 3n - 6. No issues were found in this section. We move on!

Step 2: Isolating the Variable 'n'

  • Equation from Step 1: 2n - 9 = 3n - 6
  • Step 2: 2n - 9 = 3n - 6 -2n -2n ------------------ -9 = 5n - 6

Here, the goal is to get all the 'n' terms on one side of the equation. The step begins by attempting to subtract 2n from both sides. When we subtract 2n from 2n, we are left with 0. However, the resulting equation -9 = 5n - 6 is incorrect. Subtracting 2n from the right side, 3n, should leave us with n, not 5n. The right-hand side, instead of becoming n - 6, inexplicably becomes 5n - 6. That's where the error lies, people! This is a crucial area to concentrate on since making a mistake in this area will completely throw off the rest of the problem.

Let's get it straight: it should have been n - 6 and not 5n - 6.

The error lies in Step 2.

Step 3: Continuing the Isolation

  • Equation from Step 2: -9 = 5n - 6 (Incorrect equation)
  • Step 3: -9 = 5n - 6 +6 +6 --------------- -3 = 5n

In this step, the goal is to isolate the term containing 'n' further by adding 6 to both sides of the equation. If we were working with the incorrect equation from Step 2, this step would be correctly executed, as adding 6 to both sides would indeed result in -3 = 5n. But, because Step 2 has an error, so this step is built on a faulty foundation. While the arithmetic here is correct based on the erroneous equation from the previous step, it's irrelevant to the original problem.

This step tries to isolate the variable 'n' further, but because it relies on the results of the incorrect Step 2, this section of the solution is also incorrect. Always check each step to make sure that your answer is accurate.

Step 4: Solving for 'n'

  • Equation from Step 3: -3 = 5n (Incorrect equation)
  • Step 4: -3 / 5 = 5n / 5 n = -3/5

Here, the equation is divided by 5 to solve for 'n'. Again, if we were working with the (incorrect) equation -3 = 5n, the solution of n = -3/5 would be the right calculation, but since we are working from the original faulty step, our solution is incorrect, even though the calculation is right. The value of 'n' should be a different number since it will be found when we fix the error from Step 2.

*This step correctly solves for 'n' based on the incorrect equation derived from the previous steps. Therefore, this step is also incorrect in the context of the original equation.

Identifying the Culprit: The Error Uncovered

  • The Error: Step 2, where the equation incorrectly simplifies after subtracting 2n from both sides.
  • Correcting the Error: The equation in Step 2 should be 2n - 9 = 3n - 6. Subtracting 2n from both sides should result in -9 = n - 6.

So, after checking all the steps, Step 2 is the one that contains an error. After we figured out where the mistake occurred, let's fix it and solve the problem. Let's do a quick recap and solve the problem:

  1. Original Equation: 2n - 9 = 3(n - 2)
  2. Step 1: 2n - 9 = 3n - 6 (Correct)
  3. Step 2: -9 = n - 6 (Subtract 2n from both sides)
  4. Step 3: -3 = n (Add 6 to both sides)

Therefore, the correct answer is n = -3. Great job, everyone! We successfully identified and corrected the error. Keeping an eye on detail and meticulous checking are crucial to your mathematics journey.