Solving For X In 3x = 6x - 2 A Step By Step Guide

Introduction

Hey guys! Let's dive into solving a simple algebraic equation today. We're going to tackle the equation 3x = 6x - 2. Don't worry; it's easier than it looks! Understanding how to solve for x is a fundamental skill in algebra, and it's super useful in many areas of math and even in everyday life. Whether you're a student just starting out with algebra or someone looking to refresh your skills, this guide will walk you through each step clearly and concisely. We'll break down the equation, explain the logic behind each operation, and make sure you understand not just how to get the answer, but why the answer is what it is. By the end of this article, you'll be able to confidently solve similar equations and impress your friends with your math skills! Remember, math isn't about memorizing steps; it's about understanding the underlying concepts. So, grab a pencil and paper, and let's get started on this mathematical adventure together! We'll go through the steps one by one, making sure everything is crystal clear. No jargon, just plain English (and a little math, of course!). So, let’s embark on this journey to unravel the mystery of 'x' in our equation. Remember, every equation is like a puzzle, and solving for 'x' is like finding the missing piece. It's a rewarding process that builds your problem-solving skills and boosts your confidence in math. So, let's jump in and conquer this equation together!

Understanding the Equation: 3x = 6x - 2

Before we jump into the solution, let's break down what this equation, 3x = 6x - 2, actually means. In algebra, an equation is like a balanced scale. The equals sign (=) indicates that whatever is on the left side of the equation has the same value as what's on the right side. Our goal is to find the value of 'x' that makes this balance true. Think of 'x' as an unknown quantity – our mystery variable. The numbers in front of 'x' (like the 3 and 6) are called coefficients. They tell us how many 'x's we have. So, 3x means we have three 'x's, and 6x means we have six 'x's. The minus 2 (-2) on the right side is a constant – a value that doesn't change. It's just a plain old number hanging out there. Now, let’s rephrase the equation in simpler terms. We have three times some number 'x' on one side. On the other side, we have six times that same number 'x', but we're taking away 2. Our job is to figure out what that number 'x' is. To do this, we need to isolate 'x' on one side of the equation. This means we want to get 'x' all by itself, with a coefficient of 1 (or no coefficient at all, which is the same thing), so we can see its true value. We'll do this by performing the same operations on both sides of the equation, keeping the balance intact. Remember that balanced scale? We need to make sure we add or subtract the same amount from both sides, or we'll throw it off. So, that’s the lay of the land. We understand what the equation means, we know what our goal is (to isolate 'x'), and we have a general idea of how we're going to get there (by keeping the equation balanced). Now, let’s dive into the step-by-step solution.

Step-by-Step Solution

Alright, let's get down to business and solve this equation 3x = 6x - 2 step by step. Our main goal here is to isolate 'x' on one side of the equation. This means we want to manipulate the equation until we have 'x' by itself on either the left or right side. Think of it like peeling an onion – we need to carefully remove layers until we get to the core. The first thing we want to do is gather all the terms with 'x' on one side of the equation. Currently, we have 3x on the left and 6x on the right. To get them together, we can subtract 6x from both sides. Why 6x? Because it will eliminate the 6x term on the right side. Remember, whatever we do to one side, we must do to the other to keep the equation balanced. So, let's do it:

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

Now, let’s simplify both sides. On the left, 3x - 6x combines to give us -3x. On the right, 6x - 6x cancels out, leaving us with just -2. So, our equation now looks like this:

-3x = -2

We're getting closer! We now have all the 'x' terms on the left and the constant term on the right. But we're not quite there yet. We need to get 'x' by itself, and right now, it's being multiplied by -3. To undo this multiplication, we need to divide both sides of the equation by -3. Again, we're doing the same thing to both sides to maintain that balance. Here's how it looks:

-3x / -3 = -2 / -3

Now, let's simplify again. On the left, -3x / -3 simplifies to just 'x', which is exactly what we wanted! On the right, -2 / -3 simplifies to 2/3. Remember, a negative divided by a negative is a positive. So, our final answer is:

x = 2/3

And there you have it! We've successfully solved for 'x'. We started with the equation 3x = 6x - 2, and through a series of careful steps, we've found that x is equal to 2/3.

Verification of the Solution

Now that we've found our solution, x = 2/3, it's always a good idea to double-check our work. This is called verification, and it's a crucial step in problem-solving. It ensures that our answer is correct and that we haven't made any mistakes along the way. To verify our solution, we'll plug x = 2/3 back into the original equation, 3x = 6x - 2. If both sides of the equation are equal after we substitute, then we know our solution is correct. Let's start by substituting 2/3 for 'x' in the left side of the equation, which is 3x:

3 * (2/3)

When we multiply 3 by 2/3, the 3s cancel out, leaving us with just 2. So, the left side of the equation simplifies to 2. Now, let's do the same for the right side of the equation, which is 6x - 2. We'll substitute 2/3 for 'x' again:

6 * (2/3) - 2

First, we multiply 6 by 2/3. This gives us 12/3, which simplifies to 4. So, our expression now looks like this:

4 - 2

Subtracting 2 from 4, we get 2. So, the right side of the equation also simplifies to 2. Now, let's compare the two sides. The left side simplified to 2, and the right side simplified to 2. Since both sides are equal, we can confidently say that our solution, x = 2/3, is correct! Verification is a powerful tool. It not only confirms your answer but also reinforces your understanding of the problem-solving process. It’s like having a built-in error detector. If the two sides of the equation don't match after substitution, you know you need to go back and check your steps. So, always remember to verify your solutions, especially in exams or when dealing with complex problems. It's a small step that can make a big difference in your accuracy and confidence.

Real-World Applications

Solving equations like 3x = 6x - 2 might seem like an abstract mathematical exercise, but it actually has tons of real-world applications. Algebra, in general, is a powerful tool for modeling and solving problems in various fields. Let's explore a few examples where solving for 'x' can be incredibly useful. Imagine you're planning a budget. You know you want to save a certain amount of money each month, but you also have expenses to consider. You could use an equation to figure out how much you need to earn to meet your savings goal. For example, let's say you want to save $100 per month (x), and you have fixed expenses of $500. If you earn $700, you can set up an equation like 700 = x + 500 and solve for x to see how much you can save. In science, equations are used to describe relationships between different variables. For instance, in physics, you might use an equation to calculate the distance an object travels based on its speed and time. Solving for 'x' in this context could help you determine how long it will take to reach a destination or how fast you need to travel to cover a certain distance. In engineering, algebraic equations are used in designing structures, circuits, and many other things. Engineers need to calculate forces, stresses, and currents, and they often need to solve for unknown variables to ensure their designs are safe and efficient. Even in everyday situations like cooking, you might use equations to scale recipes up or down. If a recipe calls for a certain amount of ingredients for four servings, but you want to make it for six, you can use algebra to figure out how much of each ingredient you need. The ability to solve for 'x' is a valuable skill that extends far beyond the classroom. It empowers you to analyze situations, make informed decisions, and solve problems in a systematic way. So, the next time you're faced with a real-world challenge, remember that algebra might just be the tool you need to find the solution.

Conclusion

So, there you have it! We've successfully solved the equation 3x = 6x - 2, and we've found that x = 2/3. We walked through the process step by step, from understanding the equation to verifying our solution. We also explored some real-world applications of solving for 'x', showing how this skill is relevant in various fields and everyday situations. Remember, solving algebraic equations is like building a muscle – the more you practice, the stronger you become. Don't be afraid to tackle different types of equations and challenge yourself. The key is to understand the underlying principles and apply them consistently. Always remember to keep the equation balanced by performing the same operations on both sides. And always, always verify your solution to make sure you're on the right track. Algebra is a foundational skill in mathematics, and mastering it opens doors to more advanced concepts and problem-solving techniques. Whether you're pursuing a career in science, engineering, finance, or any other field that involves quantitative analysis, a solid understanding of algebra will serve you well. But beyond its practical applications, algebra also develops critical thinking skills, logical reasoning, and the ability to approach problems systematically. It teaches you to break down complex situations into smaller, manageable parts and to identify patterns and relationships. So, keep practicing, keep exploring, and keep solving! Math can be challenging, but it's also incredibly rewarding. And with each equation you solve, you're not just finding a numerical answer; you're building your problem-solving skills and your confidence in your abilities. You've got this, guys! Keep up the great work, and happy solving!