Error In Equation Solving: Can You Spot It?

Hey guys! Ever feel like you're acing an equation, only to find out you've made a tiny little mistake that throws everything off? It happens to the best of us! Today, we're going to dive deep into a solved equation and play detective to find the mistake. This is a super important skill in mathematics because it’s not just about getting the right answer; it’s about understanding the process and knowing where things can go wrong. So, let's put on our detective hats and get started!

The Case of the Mis-Solved Equation

Here's the equation and the steps taken to solve it:

$\begin{aligned} 6 x-1 & =-2 x+9 \\ 8 x-1 & =9 \\ 8 x & =10 \\ x & =\frac{8}{10} \\ x & =\frac{4}{5} \end{aligned}$

Our mission, should we choose to accept it (and we do!), is to meticulously examine each step and pinpoint where the error occurred. Think of it like a puzzle – each step is a piece, and one of them doesn't quite fit. To do this effectively, we need to understand the fundamental principles that govern equation solving. We're talking about the properties of equality, inverse operations, and the order of operations. If we have a solid grasp of these concepts, spotting the mistake will be a piece of cake… or maybe a slice of pi? (Pun intended!). So, before we jump to conclusions, let's refresh our understanding of these crucial mathematical principles. Remember, a strong foundation is key to successful problem-solving. Keep your eyes peeled, and let's crack this case!

Step-by-Step Investigation: Spotting the Flaw

Let's break down each step of the equation solving process and see if we can find the culprit. We’ll go through each line, explain what was done, and check if it follows the rules of algebra. This is where we really get into the nitty-gritty, so pay close attention!

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

This is our starting point, the original equation. No operations have been performed yet, so there's no possibility of error here. It's simply the equation we need to solve. We need to isolate 'x' on one side of the equation, and to do that, we'll need to use inverse operations and the properties of equality.

Step 2: $8x - 1 = 9$

Here's where things get interesting! To get from Step 1 to Step 2, it looks like someone added $2x$ to both sides of the equation. This is a perfectly valid move, based on the Addition Property of Equality, which states that you can add the same value to both sides of an equation without changing its solution. So, $6x + 2x$ does indeed equal $8x$, and the $-2x + 2x$ on the right side cancels out, leaving us with just the 9. So far, so good! But remember, we're looking for a mistake, so we need to keep our guard up and carefully analyze the remaining steps.

Step 3: $8x = 10$

In this step, it appears that 1 was added to both sides of the equation. Again, this aligns with the Addition Property of Equality. Adding 1 to both sides of $8x - 1 = 9$ should indeed give us $8x = 10$. We're still on track, and the logic seems sound. It’s like we’re following a trail of breadcrumbs, and so far, each crumb has led us in the right direction. But don't get complacent! The mistake is lurking somewhere, waiting to be discovered.

Step 4: $x = \frac{8}{10}$

Okay, this is where our detective senses should be tingling! To isolate $x$, we need to divide both sides of the equation by the coefficient of $x$, which is 8. So, if we divide both sides of $8x = 10$ by 8, we should get $x = \frac{10}{8}$, not $x = \frac{8}{10}$. Aha! We've found our culprit! This step incorrectly divided 8 by 10 instead of 10 by 8. This is a classic mistake, often caused by rushing or not paying close attention to the operation being performed. Remember, in algebra, order matters! Dividing by a number is not the same as dividing the number by it.

Step 5: $x = \frac{4}{5}$

This step simplifies the fraction $\frac{8}{10}$ to $\frac{4}{5}$. The simplification itself is correct – both the numerator and denominator were divided by their greatest common factor, which is 2. However, since the previous step was incorrect, this answer is also incorrect, even though the simplification was done right. It's a perfect example of how one small mistake can snowball and affect the final result. So, while the arithmetic in this step is sound, it's building upon a faulty foundation.

The Smoking Gun: Pinpointing the Error

Alright, guys, we did it! We successfully identified the mistake in the equation-solving process. The error occurred in Step 4, where both sides of the equation were incorrectly divided. Instead of dividing 10 by 8, the solution incorrectly divided 8 by 10. This seemingly small error led to an incorrect value for x. Remember, it's crucial to perform the correct operation on both sides of the equation to maintain balance and arrive at the accurate solution. This reinforces the importance of carefully reviewing each step and double-checking our work, especially when dealing with fractions and division. A little extra caution can save us from making these common algebraic blunders!

Why This Matters: The Importance of Precision

Now, you might be thinking,