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

Hey guys! Let's dive into solving a simple yet fundamental algebraic equation. In this article, we're going to break down the steps to solve for x in the equation 3x = 6x - 2. This is a classic problem that reinforces key concepts in algebra, and by the end of this guide, you’ll be a pro at tackling similar problems. We’ll cover each step in detail, ensuring you understand the logic behind the process. So, grab your pencils, and let’s get started!

Understanding the Basics of Algebraic Equations

Before we jump into solving our specific equation, let's quickly review the basic principles of algebraic equations. At its core, an algebraic equation is a statement that two expressions are equal. These expressions can involve numbers, variables (like our x), and mathematical operations. The main goal when solving an equation is to isolate the variable on one side, effectively finding the value of x that makes the equation true. To achieve this, we use properties of equality, which allow us to perform the same operations on both sides of the equation without changing its balance. Common operations include addition, subtraction, multiplication, and division. Remember, the golden rule is to maintain balance – whatever you do to one side, you must do to the other. This ensures that the equation remains valid throughout the solving process. Understanding these basics is crucial because they form the foundation for solving more complex equations later on. So, let's keep these principles in mind as we move forward and start cracking our equation.

The Importance of Isolating the Variable

In solving algebraic equations, the heart of the matter really boils down to isolating that variable. Why is this so crucial, you ask? Well, when we isolate x (or whatever variable we're dealing with), we're essentially stripping away all the other numbers and operations that are cluttering up its true value. Think of it like finding the core of an onion – you peel away the layers to reveal what's truly inside. By isolating x, we're getting to the very number that makes the equation balance perfectly. It’s like finding the missing piece of a puzzle; once we know the value of x, we can confidently say we've solved the equation. This process isn’t just about getting a number; it's about understanding how each part of the equation interacts with x and how to manipulate those parts to reveal its value. So, keep your focus on isolating that variable – it’s the key to unlocking the solution and mastering algebra!

Properties of Equality: The Rules of the Game

When we talk about solving equations, we can't overlook the properties of equality; these are our rulebook in the game of algebra. These properties allow us to manipulate equations while ensuring they remain balanced and true. There are a few key players here: the Addition Property, which states that adding the same number to both sides of an equation doesn't change its validity; the Subtraction Property, which is the flip side of addition, allowing us to subtract the same number from both sides; the Multiplication Property, which says we can multiply both sides by the same number; and the Division Property, which, like subtraction, is the inverse of multiplication. These properties are the backbone of our equation-solving strategy. They give us the green light to rearrange and simplify equations without fear of messing things up. By understanding and applying these properties, we can confidently move terms around, clear out unwanted numbers, and inch closer to isolating our variable. Think of these properties as your superpowers in the world of algebra – use them wisely!

Step-by-Step Solution for 3x = 6x - 2

Alright, let's get our hands dirty and solve the equation 3x = 6x - 2. We'll break it down into manageable steps to make sure everyone's on the same page. First things first, we need to gather all the terms with x on one side of the equation. This is a crucial step in isolating our variable. Currently, we have 3x on the left and 6x on the right. To consolidate these, we can subtract 6x from both sides. Remember, whatever we do to one side, we must do to the other to keep the equation balanced. So, 3x - 6x = 6x - 6x - 2 simplifies to -3x = -2. Now we’re getting somewhere! We've managed to get all the x terms on one side. Next up, we need to get x completely by itself. It’s currently being multiplied by -3, so to undo that, we'll divide both sides by -3. This gives us (-3x) / -3 = -2 / -3, which simplifies to x = 2/3. And there we have it! We've successfully solved for x. Now, let's walk through each of these steps again, just to make sure everything clicks.

Step 1: Grouping Like Terms

The first big move in solving 3x = 6x - 2 is grouping like terms. What does this mean, exactly? Well, we want to bring all the terms that contain our variable, x, to one side of the equation. This is a fundamental strategy in algebra because it helps us simplify the equation and move closer to isolating x. In our case, we have 3x on the left side and 6x on the right side. To consolidate these, we need to decide which side we want our x terms to reside on. A common approach is to move the term with the smaller coefficient (in this case, 3x) to the side with the larger coefficient (6x). However, for this example, let's move the 6x to the left side. We can do this by subtracting 6x from both sides of the equation. This is where the properties of equality come into play. By subtracting the same quantity from both sides, we maintain the equation's balance. So, the equation becomes 3x - 6x = 6x - 6x - 2. Simplifying this gives us -3x = -2. See how we've grouped the x terms on the left? This sets us up perfectly for the next step, where we'll isolate x completely.

Step 2: Isolating x

Having grouped our like terms, we're now at the crucial step of isolating x. Remember, our goal is to get x all by itself on one side of the equation. Looking at our current equation, -3x = -2, we see that x is being multiplied by -3. To undo this multiplication and free x, we need to perform the inverse operation, which is division. We'll divide both sides of the equation by -3. Again, the golden rule of algebra applies: what we do to one side, we must do to the other. So, we divide both -3x and -2 by -3. This gives us (-3x) / -3 = -2 / -3. On the left side, the -3s cancel each other out, leaving us with just x. On the right side, we have -2 divided by -3. A negative number divided by a negative number results in a positive number. Therefore, -2 / -3 simplifies to 2/3. So, our equation now reads x = 2/3. And just like that, we've isolated x! We've found the value that makes the original equation true. But before we celebrate, let's take the extra step of verifying our solution.

Step 3: Verifying the Solution

We've solved for x, but how can we be absolutely sure our answer is correct? This is where the crucial step of verifying the solution comes into play. It's like double-checking your work to catch any sneaky errors. To verify our solution, we'll take the value we found for x, which is 2/3, and plug it back into the original equation: 3x = 6x - 2. We're essentially asking,