Equation Error: Spotting Mistakes In Math Solutions

Aug 4, 2025 by Viktoria Ivanova 52 views

Unraveling Sharon's Equation Error: A Step-by-Step Breakdown

Hey guys! Let's dive into a common algebra problem today. We're going to analyze how Sharon attempted to solve an equation and pinpoint where she went wrong. It's super common to make little mistakes in math, and that's totally okay! The important thing is learning how to identify and fix those errors. So, let's get started and break down each step of Sharon's work.

Sharon's Attempted Solution

Here's the equation Sharon was trying to solve, along with her step-by-step solution:

9 = -3(e - 2)
9 = -3e + 6  --- Step 1
15 = 3e  --- Step 2
5 = e  --- Step 3

At first glance, it might seem like Sharon is on the right track. She's applying the distributive property, isolating the variable, and solving for 'e.' However, there's a sneaky little error hiding in one of these steps. Let's meticulously analyze each step to uncover the culprit.

Step-by-Step Analysis: Spotting the Mistake

Step 1: Distribution

In Step 1, Sharon expands the expression on the right side of the equation. She distributes the -3 across the parentheses: -3 multiplied by 'e' gives -3e, and -3 multiplied by -2 gives +6. So, the equation transforms from 9 = -3(e - 2) to 9 = -3e + 6. This step is absolutely correct! Sharon has flawlessly applied the distributive property. Give yourself a pat on the back, Sharon – you nailed this one!

Understanding the distributive property is crucial in algebra. It's the foundation for simplifying expressions and solving equations. So, if you ever feel shaky on this concept, it's definitely worth revisiting. Remember, it's all about multiplying the term outside the parentheses by each term inside.

Step 2: Isolating the Variable

Now, Step 2 is where things get interesting. Sharon attempts to isolate the term with 'e' by manipulating the equation. She transitions from 9 = -3e + 6 to 15 = 3e. Let's examine this closely. To isolate the '-3e' term, we need to get rid of the '+6' on the right side. The correct way to do this is to subtract 6 from both sides of the equation. This maintains the balance and keeps the equation true.

However, Sharon seems to have made a mistake here. It looks like she might have added 6 to the left side (9 + 6 = 15), which is correct for that side, but then she incorrectly changed '-3e' to '3e'. This is where the error lies! Remember, we only subtracted 6 from both sides. The '-3e' term should remain unchanged at this point.

The correct transformation should have been:

9 - 6 = -3e + 6 - 6
3 = -3e

This highlights a critical concept in solving equations: performing the same operation on both sides to maintain equality. It's like a balancing scale – whatever you do to one side, you must do to the other to keep it level.

Step 3: Solving for 'e'

Step 3 builds upon the (incorrect) result from Step 2. Sharon divides both sides of the equation 15 = 3e by 3, resulting in 5 = e. While the division itself is performed correctly, it's based on the faulty equation from Step 2. Therefore, the final answer is also incorrect. The lesson here is that even if your final steps are accurate, a mistake earlier in the process will propagate through and lead to a wrong solution.

Identifying Sharon's Mistake: The Critical Error

So, after carefully dissecting each step, we've pinpointed Sharon's mistake. The error occurs in Step 2 when she incorrectly transforms 9 = -3e + 6 into 15 = 3e. The proper transformation should have been 3 = -3e. This mistake likely stems from a misunderstanding of how to isolate terms in an equation while maintaining balance. It's a classic algebraic pitfall, and we've all been there! The key is to remember to perform the same operation on both sides of the equation.

To reiterate, the mistake is in Step 2 where Sharon incorrectly changed the equation to 15 = 3e instead of 3 = -3e. This single error throws off the rest of the solution.

Correcting the Solution: Let's Do It Right!

Now that we've identified the mistake, let's walk through the correct solution step-by-step:

Original Equation: 9 = -3(e - 2)
Step 1: Distribute: 9 = -3e + 6 (This step was correct!)
Step 2: Isolate the 'e' term: Subtract 6 from both sides:
```
9 - 6 = -3e + 6 - 6
3 = -3e
```
Step 3: Solve for 'e': Divide both sides by -3:
```
3 / -3 = -3e / -3
-1 = e
```

Therefore, the correct solution is e = -1. See how a small error can lead to a completely different answer? That's why it's so crucial to be meticulous and double-check your work, especially in algebra!

Key Takeaways: Lessons Learned from Sharon's Mistake

Sharon's attempt, though not entirely successful, provides us with some valuable learning opportunities. Here are the key takeaways from this exercise:

The Importance of the Distributive Property: Sharon correctly applied the distributive property in Step 1, which is a fundamental skill in algebra. Mastering this property is crucial for simplifying expressions and solving equations. Remember to distribute the term outside the parentheses to every term inside.
Maintaining Balance in Equations: The core of solving equations lies in maintaining balance. Whatever operation you perform on one side of the equation, you must perform on the other side. This ensures that the equality remains true. Think of it as a mathematical seesaw – you need to keep both sides level.
Careful Isolation of Variables: Isolating the variable you're trying to solve for is a critical step. This often involves adding, subtracting, multiplying, or dividing terms on both sides of the equation. Pay close attention to the signs (positive and negative) when manipulating terms.
Double-Checking Your Work: Math errors are common, and even the most experienced mathematicians make them sometimes. That's why double-checking your work is essential. Review each step carefully and make sure it logically follows from the previous step. If possible, substitute your solution back into the original equation to verify that it holds true. This is a foolproof way to catch mistakes.
Understanding the Impact of Errors: As we saw in Sharon's case, a single error in an early step can throw off the entire solution. This underscores the importance of accuracy and attention to detail in mathematics. Even small mistakes can have significant consequences.

Practice Makes Perfect: Sharpening Your Algebraic Skills

Solving equations is a skill that improves with practice. The more you work through problems, the more comfortable and confident you'll become. Don't be discouraged by mistakes – they're simply opportunities to learn and grow. Here are a few tips for honing your algebraic prowess:

Work Through a Variety of Problems: Expose yourself to different types of equations, from simple one-step equations to more complex multi-step problems. This will help you develop a versatile toolkit of problem-solving strategies.
Break Down Complex Problems: When faced with a challenging equation, break it down into smaller, more manageable steps. This makes the problem less daunting and reduces the likelihood of errors.
Show Your Work: It's tempting to skip steps and do calculations in your head, but showing your work is crucial. It allows you to track your progress, identify potential errors, and communicate your reasoning clearly.
Seek Help When Needed: Don't hesitate to ask for help if you're struggling with a concept or a particular problem. Talk to your teacher, classmates, or a tutor. There's no shame in seeking assistance, and it can make a big difference in your understanding.
Utilize Online Resources: The internet is a treasure trove of math resources, including tutorials, practice problems, and interactive tools. Explore websites like Khan Academy, Wolfram Alpha, and Mathway to supplement your learning.

By understanding the concepts, practicing diligently, and learning from your mistakes, you can master the art of solving equations and excel in algebra.

Conclusion: Embracing the Learning Process

Sharon's equation-solving journey highlights the importance of precision and attention to detail in algebra. While she made a mistake in Step 2, the process of identifying and correcting that error provides a valuable learning experience. Math isn't always about getting the right answer on the first try; it's about understanding the concepts, analyzing your work, and persevering through challenges. So, keep practicing, stay curious, and embrace the learning process! You've got this, guys!