Spot The Error In This Algebra Equation

Oct 23, 2025 by Dimemap Team 40 views

Hey guys, let's dive into a common scenario where we're trying to solve an algebraic equation, and sometimes, even the pros can slip up! Today, we've got a step-by-step solution to an equation, and our mission, should we choose to accept it, is to find the exact mistake made along the way. This isn't just about getting the right answer; it's about understanding the process and recognizing where things can go sideways. So, grab your thinking caps, and let's get this done!

The Equation and the Solution Steps

First off, let's lay out the equation and the proposed steps for solving it. The original equation is: $\begin{aligned} 6 x-1 & =-2 x+9 \\ 8 x-1 & =9 \\ 8 x & =10 \\ x & =\frac{8}{10} \\ x & =\frac{4}{5} \\end{aligned}$

Our goal is to meticulously examine each step, from the very first move to the final simplified answer, to pinpoint where the error occurred. Think of it like a detective case – we need to follow the clues (the mathematical operations) and see if they lead us logically from one step to the next. Sometimes, the mistake is super obvious once you spot it, but other times, it's a sneaky little slip-up that can be hard to find if you're not paying close attention.

Step 1: Combining Like Terms

The first step shown is transitioning from $6x - 1 = -2x + 9$ to $8x - 1 = 9$ . Let's break down what's happening here. To get from the first line to the second, it looks like someone decided to combine the 'x' terms. On the left side, we have $6x$ , and on the right side, we have $-2x$ . To get all the 'x' terms onto one side, a common strategy is to add $2x$ to both sides of the equation. If we do that:

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

This simplifies to:

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

$8x - 1 = 0 + 9$

$8x - 1 = 9$

Boom! This first step appears to be perfectly executed. The 'x' terms have been successfully gathered on the left side, and the constants remain on the right. This is a crucial move in isolating the variable, and it seems like it was done correctly. So, if you were thinking the error might be here, you can relax – this part is solid.

Step 2: Isolating the Variable Term

Now, we move to the second step, which takes us from $8x - 1 = 9$ to $8x = 10$ . Let's analyze this transition. Our aim in this step is usually to get the term with the variable (in this case, $8x$ ) all by itself on one side of the equation. We currently have '-1' on the same side as $8x$ . To eliminate that '-1', we need to perform the opposite operation, which is adding 1. And, crucially, whatever we do to one side of the equation, we must do to the other side to maintain balance. So, let's add 1 to both sides of $8x - 1 = 9$ :

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

This simplifies to:

$8x + 0 = 10$

$8x = 10$

Hold on a second! Let's re-examine the original solution provided. It states that the step from $8x - 1 = 9$ leads to $8x = 10$ . Our calculation shows the exact same result. This means that Step 2, as presented in the problem's solution, is actually correct. The mistake must lie further down the line, or perhaps one of the options provided is misleading. Let's keep going and see if we can find a discrepancy later on, or if we need to reconsider the options.

Step 3: Solving for x

Alright, moving on to the next step. We're starting with $8x = 10$ , and the next line in the provided solution is $x = rac{8}{10}$ . To solve for $x$ when we have $8x = 10$ , we need to undo the multiplication of $x$ by 8. The opposite of multiplying by 8 is dividing by 8. So, we divide both sides of the equation by 8:

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

This simplifies to:

$1x = \frac{10}{8}$

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

Now, let's compare this to the solution's third step: $x = rac{8}{10}$ . Our calculated result is $x = rac{10}{8}$ . These are not the same! The solution seems to have flipped the numerator and the denominator. Instead of dividing 10 by 8, it looks like it divided 8 by 10. This is a major red flag, guys! This is precisely where the calculation went wrong.

Step 4: Simplifying the Fraction

The solution then proceeds from $x = rac{8}{10}$ to $x = rac{4}{5}$ . This step involves simplifying the fraction $rac{8}{10}$ . To simplify a fraction, we find the greatest common divisor (GCD) of the numerator and the denominator and divide both by it. The GCD of 8 and 10 is 2. So, if we divide both the numerator and the denominator by 2:

$\frac{8 \div 2}{10 \div 2} = \frac{4}{5}$

This simplification step is mathematically correct. The fraction $rac{8}{10}$ does indeed simplify to $rac{4}{5}$ . However, because the fraction itself ( $x = rac{8}{10}$ ) was derived incorrectly in the previous step, this correct simplification leads to an incorrect final answer.

Analyzing the Options Provided

Let's revisit the options given to identify the mistake:

A. Step 2 is incorrect and should be $8x=8$ . We've already analyzed Step 2. We found that starting from $8x - 1 = 9$ and correctly performing the operation to isolate $8x$ leads to $8x = 10$ . Therefore, this option is incorrect. The statement that Step 2 should be $8x=8$ is also wrong; adding 1 to 9 correctly results in 10, not 8.
B. The mistake occurs in Step 3, where $x= rac{8}{10}$ should be $x= rac{10}{8}$ . Bingo! This perfectly matches our findings. In Step 3, when solving $8x = 10$ for $x$ , we must divide 10 by 8, not 8 by 10. The solution incorrectly wrote $x = rac{8}{10}$ when it should have been $x = rac{10}{8}$ . This is the critical error.

The Correct Solution Path

To solidify our understanding, let's walk through the correct way to solve this equation:

Start with the original equation: $\qquad 6x - 1 = -2x + 9$
Add $2x$ to both sides to combine x terms: $\qquad 6x - 1 + 2x = -2x + 9 + 2x$ $\qquad 8x - 1 = 9$ (This matches the solution's Step 1 and is correct.)
Add 1 to both sides to isolate the $8x$ term: $\qquad 8x - 1 + 1 = 9 + 1$ $\qquad 8x = 10$ (This matches the solution's Step 2 and is correct.)
Divide both sides by 8 to solve for x: $\qquad \frac{8x}{8} = \frac{10}{8}$ $\qquad x = \frac{10}{8}$ ( extbf{This is where the provided solution made its mistake. It wrote $\frac{8}{10}$ instead of $\frac{10}{8}$ .})
Simplify the fraction: $\qquad x = \frac{10 \div 2}{8 \div 2}$ $\qquad x = \frac{5}{4}$

So, the correct answer to the equation is $x = rac{5}{4}$ , not $x = rac{4}{5}$ .

Conclusion

Great job sticking with it, guys! We've successfully identified the mistake in the provided steps. The error occurred in Step 3, where the division was performed incorrectly, leading to $x = rac{8}{10}$ instead of the correct $x = rac{10}{8}$ . Even though the subsequent simplification step was mathematically sound, it was applied to an incorrect value. Remember, in mathematics, every single step counts, and a small error early on can snowball into a completely wrong final answer. Keep practicing, stay sharp, and you'll be spotting these kinds of mistakes like a pro in no time! This is a classic example of how important it is to double-check your work, especially when it comes to understanding which number goes in the numerator and which goes in the denominator after division. Keep up the great work!