Solving For X: Your Step-by-Step Guide

Hey guys, welcome back to the channel! Today, we've got a super fun algebra problem for you: Solve for x. We're diving deep into the world of equations, specifically tackling this one: 3x = 6x - 2. If you're into math, or even if you just need a refresher on how to isolate a variable, you've come to the right place! We're going to break this down step-by-step, making it as clear as possible. So, grab your notebooks, maybe a cup of coffee, and let's get our algebra hats on. Our main goal here is to figure out what numerical value 'x' represents in this equation. It might seem simple to some, but understanding the fundamental principles behind solving for 'x' is absolutely crucial for more complex mathematical concepts down the line. Think of it like learning the alphabet before you can write a novel – you've gotta nail the basics! We'll explore the common pitfalls people fall into and how to avoid them, ensuring you walk away feeling confident and ready to tackle similar problems. We're not just going to give you the answer; we're going to teach you how to get there, every single time. So, stick around, and let's make solving for 'x' a breeze!

Understanding the Equation: 3x = 6x - 2

Alright, let's get down to business with our equation: 3x = 6x - 2. What we're essentially trying to do here, guys, is to isolate the variable 'x' on one side of the equals sign. Think of the equals sign as a balance scale. Whatever you do to one side, you have to do to the other to keep it balanced. Our mission, should we choose to accept it, is to get 'x' all by itself. Right now, 'x' is chilling on both sides of the equation, and it's making a mess. We've got '3x' on the left and '6x' on the right, along with that pesky '- 2' hanging out with the '6x'. To solve for 'x', we need to gather all the 'x' terms together on one side and all the constant numbers (like our '- 2') on the other. It’s a bit like tidying up your room – you put all the clothes in one pile and all the books in another. The process involves using inverse operations. For instance, if you see a '+ 5', you'd use a '- 5' to get rid of it. If you see a 'multiplied by 2', you'd use 'divided by 2'. We'll be applying these exact principles here. The goal is to manipulate the equation using these inverse operations without changing its fundamental truth, which is that the left side always equals the right side. This is the golden rule of algebra, and if you remember this, you're halfway to solving any equation. So, our first step is going to be to decide where we want to move our 'x' terms. It doesn't really matter which side you choose, but sometimes it's easier to move the smaller 'x' term to avoid dealing with negative coefficients for 'x' initially, though that's not always the case. We'll talk about that strategy in a moment. But first, let's visualize this equation. On one side, we have three times some unknown number 'x'. On the other side, we have six times that same number 'x', minus two. We need to find that specific number 'x' that makes this statement true. It's like a mystery waiting to be solved, and the clues are all laid out in the equation itself.

Step-by-Step Solution: Isolating 'x'

Okay, team, let's get our hands dirty and actually solve this thing! We have 3x = 6x - 2. Remember our goal: get 'x' by itself. The first thing we want to do is get all the 'x' terms onto one side. I usually prefer to move the 'x' term that will result in a positive coefficient for 'x', but either way works. Let's subtract '3x' from both sides of the equation. Why? Because we want to eliminate the '3x' on the left side. Whatever we do to one side, we must do to the other to keep the equation balanced.

So, we have:

3x - 3x = 6x - 3x - 2

On the left side, 3x - 3x equals zero. That's exactly what we wanted – to clear the left side of 'x's!

On the right side, 6x - 3x simplifies to 3x.

So now, our equation looks like this:

0 = 3x - 2

See? We've successfully moved all the 'x' terms to the right side. Now, we need to get the 'x' term completely isolated from the constant term (- 2). The '- 2' is currently attached to the 3x. To get rid of that '- 2', we need to do the opposite operation. Since it's '- 2', we're going to add '2' to both sides of the equation.

Let's do it:

0 + 2 = 3x - 2 + 2

On the left side, 0 + 2 is simply 2.

On the right side, - 2 + 2 equals zero, leaving us with just 3x.

Our equation now reads:

2 = 3x

We're so close, guys! 'x' is almost alone. It's currently being multiplied by 3. To undo multiplication, we use division. So, we need to divide both sides of the equation by 3.

2 / 3 = 3x / 3

On the left side, we just have 2/3.

On the right side, 3x / 3 simplifies to x (because the 3s cancel out).

And there you have it!

2/3 = x

So, the solution to our equation 3x = 6x - 2 is x = 2/3. We successfully isolated 'x' by using inverse operations and keeping the equation balanced at every step. Pretty neat, right?

Verifying the Solution: Plugging it Back In

Now, for the most important part, guys: checking our work! It’s always a good idea to plug our answer back into the original equation to make sure it holds true. This is how you avoid those sneaky little errors that can creep into algebra. Our original equation was 3x = 6x - 2, and we found that x = 2/3. Let's substitute 2/3 wherever we see 'x' in the original equation and see if both sides are equal.

First, let's look at the left side: 3x.

Substitute x = 2/3:

3 * (2/3)

When you multiply 3 by 2/3, the 3 in the numerator of 3/1 cancels out with the 3 in the denominator of 2/3. So, 3 * (2/3) = 2.

Now, let's look at the right side: 6x - 2.

Substitute x = 2/3:

6 * (2/3) - 2

First, let's calculate 6 * (2/3). We can think of 6 as 6/1. So, (6/1) * (2/3) = (6 * 2) / (1 * 3) = 12 / 3.

12 / 3 simplifies to 4.

So, the right side becomes 4 - 2.

And 4 - 2 equals 2.

Now, let's compare the left side and the right side:

Left side = 2

Right side = 2

Since the left side equals the right side (2 = 2), our solution x = 2/3 is correct! High five! This verification step is super crucial. It gives you confidence in your answer and helps solidify your understanding of how equations work. If you had gotten different numbers, you'd know it was time to go back and review your steps. But in this case, everything checks out perfectly!

Common Mistakes and How to Avoid Them

When you're solving for 'x', guys, there are a few common tripping hazards that can throw you off track. One of the biggest ones is sign errors. Forgetting to change the sign when you move a term to the other side of the equals sign is a classic mistake. Remember, when you add or subtract a term from one side, you must do the opposite operation on the other side, and that includes the sign. Another pitfall is arithmetic mistakes, especially when dealing with fractions or negative numbers. Double-checking your calculations, like we did in the verification step, is key. For example, if you miscalculated 6 * (2/3), you might end up with the wrong answer. Always be meticulous with your arithmetic. Also, make sure you're not mixing up your operations. If 'x' is being multiplied, you need to divide; if it's being added, you need to subtract. Don't, for example, try to subtract a number when you should be dividing. Sometimes, people get confused about which side to move the 'x' terms to. While it doesn't technically matter for the final answer, choosing the side that results in a positive coefficient for 'x' can help prevent sign errors later on. In our case, moving 3x from the left to the right (by subtracting 3x from both sides) was a good strategy because it left us with 3x on the right instead of -3x if we had moved 6x to the left. Finally, one of the most overlooked steps is not verifying your answer. Seriously, guys, this is your safety net! Always plug your solution back into the original equation. It takes just a minute and can save you a lot of frustration. If you see a mismatch, you know exactly where to look for your error. By being aware of these common mistakes and practicing consistently, you'll become a pro at solving for 'x' in no time!

Conclusion: You've Solved for x!

And there you have it, mathematicians in training! We've successfully tackled the equation 3x = 6x - 2 and found our solution: x = 2/3. We walked through the entire process, from understanding what it means to solve for x to meticulously isolating the variable using inverse operations. We made sure to keep the equation balanced at every step, just like a perfectly calibrated scale. Then, we reinforced our answer by plugging it back into the original equation, proving that our hard work paid off and our solution is indeed correct. We also discussed those common algebraic blunders – those sneaky sign errors and arithmetic slip-ups – and how to sidestep them with careful practice and diligent checking. Solving for 'x' might seem like a small step, but it’s a foundational skill in mathematics that opens doors to understanding more complex concepts, from linear equations to calculus and beyond. So, whether you're acing a test, working on homework, or just enjoy the puzzle of numbers, you've now got another tool in your algebraic toolbox. Keep practicing, keep questioning, and never be afraid to check your work. You guys are awesome, and we'll catch you in the next video! Keep exploring the wonderful world of math!