Unveiling The Error: Solving Equations Step By Step

Hey guys, let's dive into a common algebra problem! We're going to dissect a set of steps used to solve an equation and pinpoint the mistake. It's like being a detective, but instead of solving a crime, we're solving for x! Understanding where we went wrong is super important because it helps us avoid these pitfalls in the future. Let's get started!

The Equation and the Steps

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

$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}$

And here's a breakdown of the supposed reasoning behind each step:

1. Addition property of equality
2. Addition property of equality

Right off the bat, looks like a classic algebra problem. Our goal is to isolate x on one side of the equation. It looks like we've got a few steps to go through. Let's check each one and see if everything is right. Pay close attention to where the mistake might be hiding! Remember, we need to make sure each step follows the rules of algebra. We'll look at the original equation and then follow the steps and see what happens, always double-checking our work to ensure everything adds up, you know? Because, sometimes, even the smallest errors can lead us down the wrong path. Now, let's get into the meat of the matter, shall we?

Step-by-Step Analysis: Spotting the Error

Alright, let's get our detective hats on and go through these steps one by one. We need to be super careful and make sure each move is mathematically sound.

Step 1: Applying the Addition Property of Equality

• Original Equation: $6x - 1 = -2x + 9$

Looking at the first step, it seems like the goal was to get all the x terms on one side of the equation. The addition property of equality states that if we add the same value to both sides of an equation, the equation remains balanced. In the correct first step, we need to add $2x$ to both sides of the equation and add 1 to both sides. Let's show our steps.

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

Which simplifies to:

$8x = 10$

• The Incorrect Step: $8x - 1 = 9$

So what the user did here was add $2x$ to both sides of the equation, they seemed to have missed adding $2x$ to $6x$, instead they just wrote down $8x$. They also have to add $1$ to both sides of the equation, and that seems to be what the user wanted to do in the next step. The user also forgot to add $2x$ to the left side of the equation. Instead of what they wanted to do here, they should've added $2x$ to both sides first, then added $1$ to both sides. It looks like our error is here, guys. Let's keep going, just in case, though!

Step 2: Applying the Addition Property of Equality (Again)

• Previous Result (Incorrect): $8x - 1 = 9$

In this step, the user is trying to isolate x by getting rid of the $-1$ term by using the addition property of equality. However, the first step itself was wrong, so this is not the right step. They add $1$ to both sides of the equation, let's see where this step takes us.

• The Incorrect Step: $8x = 10$

Adding 1 to both sides would give $8x = 10$, which matches our result from our correct first step! Let's see if the user can solve the equation in the next step.

Step 3: Solving for x

• Previous Result: $8x = 10$

To find the value of x, we need to isolate it by dividing both sides of the equation by the coefficient of x. It's the fundamental rule of algebra. When you divide both sides of an equation by the same non-zero number, the equality is maintained. So, let's divide both sides by 8.

$8x / 8 = 10 / 8$

This simplifies to

$x = 10 / 8$

Now the user would need to simplify it to its simplest form.

• The Incorrect Step: $x = 8 / 10$

Okay, so in this step, they wrote $8$ in the numerator and $10$ in the denominator. That's not correct! The numerator and denominator are reversed! The result from the last step was $8x=10$, so they needed to divide both sides by $8$. Let's move on.

Step 4: Simplifying the Result

• Previous Result (Incorrect): $x = 8 / 10$

Now, the user has to simplify the fraction to find the value of x.

• The Incorrect Step: $x = 4 / 5$

In this step, the user simplifies the fraction from the previous step, and, thankfully, this step is correct! $x = 8/10$ becomes $x = 4/5$.

Identifying the Mistake

So, where did we go wrong? The big mistake happened in the first step. The user incorrectly combined terms and skipped adding $2x$ to both sides, and $1$ to both sides. This error caused a chain reaction, leading to an incorrect answer. It's super important to meticulously apply the addition property of equality, ensuring that we add the same value to both sides of the equation. It's all about being precise, you know?

The Correct Solution

Let's go through the correct steps to make sure we understand how to solve this equation correctly.

1. Original Equation: $6x - 1 = -2x + 9$
2. Add $2x$ to both sides: $6x - 1 + 2x = -2x + 9 + 2x$ which simplifies to $8x - 1 = 9$
3. Add $1$ to both sides: $8x - 1 + 1 = 9 + 1$ which simplifies to $8x = 10$
4. Divide both sides by $8$: $8x / 8 = 10 / 8$ which simplifies to $x = 10/8$
5. Simplify the fraction: $x = 5 / 4$

See? Now we're cooking with gas! Always double-check your work at each step. Doing this will help you avoid silly mistakes and build your confidence when solving equations.

Conclusion

Alright, guys, we've uncovered the mistake and learned from it. Remember, when solving equations, pay close attention to the rules, be patient, and always double-check your work. By following the correct steps and being careful with our math, we can conquer any algebra problem! Keep practicing, and you'll become equation-solving masters in no time. Peace out!