Equation Error: Spot The Mistake In The Solution!

Hey Plastik Magazine readers! Ever feel like you're staring at a math problem and something just isn't adding up? Well, you're not alone! Math errors happen, and sometimes they're sneaky little devils. Today, we're diving into an equation to play detective and pinpoint exactly where the mistake was made. Get your thinking caps on, guys, because we're about to unravel a mathematical mystery!

Let's Break Down the Problem

Before we jump into the solution, let's take a look at the equation we're working with. We've got a multi-step equation here, which means there are several operations we need to perform to isolate the variable and find its value. The given equation and the steps to solve it are:

$egin{aligned} 6 x-1 & =-2 x+9 \ 8 x-1 & =9 \ 8 x & =10 \ x & =rac{8}{10} \ x & =rac{4}{5}

\end{aligned}$

Now, at first glance, this might seem perfectly fine. The steps look logical, right? But remember, in math, precision is key! One tiny slip-up can throw off the entire solution. That's why we need to meticulously examine each step to make sure it follows the rules of algebra. We need to think critically about the properties of equality being applied and whether they're being used correctly. So, let's put on our detective hats and get started! We're going to go through each line, scrutinizing every operation until we find that pesky error. Are you guys ready to become equation error experts? Let's do this!

Step-by-Step Analysis: Uncovering the Flaw

Okay, guys, let's get down to business! We're going to dissect each step of the equation solution to pinpoint exactly where the mistake occurred. Remember, the key to solving equations is to maintain balance – whatever operation you perform on one side, you must perform on the other. Let’s examine the provided steps carefully.

Step 1: $6x - 1 = -2x + 9$

This is our starting point, the original equation. It looks perfectly fine. No operations have been performed yet, so there's no possibility of an error here. We've simply stated the problem. We need to keep this equation in mind as our baseline to compare against subsequent steps. It's important to ensure that each transformation we make preserves the equality and leads us closer to isolating the variable x. So far, so good!

Step 2: $8x - 1 = 9$

Now, this is where things get interesting. To get from the first equation ($6x - 1 = -2x + 9$) to this one ($8x - 1 = 9$), we need to figure out what operation was performed. It seems like someone tried to combine the x terms. To do that, we'd typically add $2x$ to both sides of the original equation. Doing so, we should get:

$6x - 1 + 2x = -2x + 9 + 2x$

Which simplifies to:

$8x - 1 = 9$

Hey, wait a minute! That's exactly what we have in the second step! So, on the surface, this step appears correct. The addition property of equality seems to have been applied correctly. But hold on a second... let’s make sure we aren't missing anything.

Step 3: $8x = 10$

Moving on to the third step, we see the equation $8x = 10$. What operation transformed $8x - 1 = 9$ into this? It looks like someone added 1 to both sides. Let's check if that's correct. If we add 1 to both sides of the equation $8x - 1 = 9$, we get:

$8x - 1 + 1 = 9 + 1$

Which simplifies to:

$8x = 10$

This step also looks perfectly valid! The addition property of equality has been correctly applied here. We're maintaining balance on both sides of the equation. The constant term has been successfully moved to the right side of the equation, bringing us closer to isolating x. This is great! We're narrowing down the possibilities for the error.

Step 4: x = rac{8}{10}

Now we're at step four: x = rac{8}{10}. To get here from $8x = 10$, we would need to isolate x by dividing both sides of the equation by 8. Remember, the division property of equality states that we can divide both sides of an equation by the same non-zero number without changing the solution. So, let's perform that operation on $8x = 10$:

rac{8x}{8} = rac{10}{8}

This simplifies to:

$x = \frac{10}{8}$

Aha! Here’s our culprit! Notice that the solution should be $x = \frac{10}{8}$, but the step shows $x = \frac{8}{10}$. It seems like someone flipped the numerator and the denominator during the division step. This is a classic algebraic mistake, and it highlights the importance of carefully applying each operation. We've found the error! The division was performed incorrectly. Instead of dividing 10 by 8, it looks like they divided 8 by 10.

Step 5: x = rac{4}{5}

Finally, we have the last step: $x = \frac{4}{5}$. This step attempts to simplify the fraction from the previous step. However, since the previous step was already incorrect, this simplification, while correct in its execution, is based on a flawed premise. If we were to simplify the correct fraction, \frac{10}{8}, we would divide both the numerator and the denominator by their greatest common divisor, which is 2. This would give us:

$\frac{10 ÷ 2}{8 ÷ 2} = \frac{5}{4}$

So, while the simplification process itself is accurate, it's applied to the wrong value. The correct simplified answer should be $x = \frac{5}{4}$, not $x = \frac{4}{5}$.

The Verdict: Error Found!

Alright, math detectives, we've cracked the case! The mistake was made in Step 4. Instead of dividing 10 by 8 to isolate x, the equation incorrectly states $x = \frac{8}{10}$. The correct step should have been $x = \frac{10}{8}$.

This highlights a crucial lesson in algebra: always double-check your work, guys! A small error in one step can lead to a completely wrong answer. By carefully examining each step and applying the properties of equality correctly, we can avoid these pitfalls. The correct solution, as we saw, simplifies to $x = \frac{5}{4}$.

Why This Matters: The Importance of Precision in Math

You might be thinking,