Solving For X The Equation 3x = 6x - 2 Explained

Hey guys! Ever stumbled upon a math problem that looks like a tangled mess of numbers and letters? Don't worry, we've all been there. Today, we're going to unravel a classic equation: 3x = 6x - 2. This might seem daunting at first glance, but trust me, it's totally solvable with a few simple steps. So, grab your pencils, open your notebooks, and let's dive into the world of algebra!

Understanding the Basics of Algebraic Equations

Before we jump into solving our specific equation, let's take a moment to grasp the fundamental principles of algebraic equations. At its core, an equation is a mathematical statement that asserts the equality of two expressions. Think of it like a balanced scale; both sides must hold the same weight for the scale to remain level. In our case, the equation 3x = 6x - 2 states that the expression on the left side (3x) is equal to the expression on the right side (6x - 2).

Now, the star of our show is the variable, denoted by the letter 'x'. Variables represent unknown quantities, and our mission is to find the value of 'x' that makes the equation true. To achieve this, we'll employ a series of algebraic manipulations, always ensuring that we maintain the balance of the equation. The key is to perform the same operations on both sides, just like adding or removing equal weights from both sides of our scale. This keeps the equation balanced and leads us closer to the solution. For instance, if we add 2 to both sides of the equation, we get 3x + 2 = 6x - 2 + 2, which simplifies to 3x + 2 = 6x. Notice how adding the same value to both sides doesn't change the fundamental equality.

We'll also use the concept of inverse operations. Inverse operations are actions that undo each other, like addition and subtraction, or multiplication and division. For example, to isolate 'x', we might need to subtract 6x from both sides or divide both sides by a coefficient of 'x'. By strategically applying inverse operations, we can gradually isolate 'x' and reveal its true value. So, with these foundational concepts in mind, let's tackle our equation and conquer the mystery of 'x'!

Step-by-Step Solution: Solving 3x = 6x - 2

Alright, let's get down to business and solve the equation 3x = 6x - 2 step by step. We'll break it down into manageable chunks, so you can follow along easily. Remember, the goal is to isolate 'x' on one side of the equation.

Step 1: Grouping the 'x' Terms

Our first move is to gather all the terms containing 'x' on one side of the equation. Currently, we have 3x on the left side and 6x on the right side. To bring them together, we can subtract 6x from both sides. This might seem like a simple step, but it's crucial for isolating 'x'. When we subtract 6x from both sides, we get:

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

Simplifying this, we combine the 'x' terms on the left side (3x - 6x = -3x) and cancel out the 6x terms on the right side (6x - 6x = 0). This leaves us with:

-3x = -2

Now, we have all the 'x' terms on the left side and a constant term on the right side. We're one step closer to isolating 'x'. This step highlights the importance of maintaining balance in the equation. Subtracting 6x from both sides ensures that the equality remains intact. Without this step, we'd have 'x' terms scattered on both sides, making it much harder to solve for 'x'. So, remember, grouping like terms is a fundamental technique in algebra.

Step 2: Isolating 'x'

We're almost there! We have -3x = -2. Now, we need to get 'x' all by itself. Currently, 'x' is being multiplied by -3. To undo this multiplication, we'll use the inverse operation: division. We'll divide both sides of the equation by -3. This will effectively isolate 'x' on the left side.

So, we divide both sides by -3:

(-3x) / -3 = (-2) / -3

On the left side, the -3 in the numerator and the -3 in the denominator cancel each other out, leaving us with just 'x'. On the right side, we have -2 divided by -3. Remember, when you divide a negative number by a negative number, the result is positive. So, -2 / -3 becomes 2/3.

Therefore, we have:

x = 2/3

And there you have it! We've successfully isolated 'x' and found its value. This step demonstrates the power of inverse operations. By dividing both sides by -3, we undid the multiplication and revealed the true value of 'x'. This technique is essential for solving a wide range of algebraic equations. So, remember, when you want to isolate a variable, think about the operations that are being applied to it and use their inverses to set it free.

Verification: Ensuring Our Solution is Correct

Before we declare victory, it's always a good idea to verify our solution. This ensures that we haven't made any mistakes along the way. To verify our solution, we'll substitute the value we found for 'x' (x = 2/3) back into the original equation (3x = 6x - 2). If both sides of the equation are equal after the substitution, then our solution is correct.

Let's substitute x = 2/3 into the left side of the equation (3x):

3 * (2/3) = 2

Now, let's substitute x = 2/3 into the right side of the equation (6x - 2):

6 * (2/3) - 2 = 4 - 2 = 2

We see that both sides of the equation are equal to 2 when x = 2/3. This confirms that our solution is correct! Verification is a crucial step in problem-solving. It's like double-checking your work to make sure you haven't made any silly errors. By substituting the solution back into the original equation, we can be confident that we've found the correct answer. So, always remember to verify your solutions, especially in algebra, where a small mistake can lead to a wrong answer.

Alternative Methods for Solving the Equation

While we've successfully solved the equation 3x = 6x - 2 using our step-by-step method, it's always beneficial to explore alternative approaches. Different methods can provide a deeper understanding of the problem and sometimes offer a more efficient solution. Let's take a look at a couple of alternative ways to tackle this equation.

Method 1: Adding 2 to Both Sides First

In our original approach, we started by grouping the 'x' terms. However, we could have chosen a different starting point. Instead of subtracting 6x from both sides, we could have added 2 to both sides first. This would eliminate the constant term (-2) on the right side. Let's see how it works:

Starting with 3x = 6x - 2, we add 2 to both sides:

3x + 2 = 6x - 2 + 2

Simplifying, we get:

3x + 2 = 6x

Now, we can subtract 3x from both sides to group the 'x' terms:

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

This simplifies to:

2 = 3x

Finally, we divide both sides by 3 to isolate 'x':

2 / 3 = x

So, we arrive at the same solution: x = 2/3. This method demonstrates that there can be multiple pathways to the same solution in algebra. The key is to choose a path that seems logical and efficient to you.

Method 2: Visual Representation (Optional)

For some people, visualizing the equation can be helpful. While this method might not be practical for complex equations, it can provide a conceptual understanding for simpler ones. Imagine a balance scale. On one side, we have 3 'x' units. On the other side, we have 6 'x' units and a weight of -2. To balance the scale, we need to figure out what value of 'x' will make both sides equal.

We can visualize subtracting 6 'x' units from both sides. This leaves us with -3 'x' units on one side and a weight of -2 on the other side. To find the value of one 'x' unit, we need to divide the weight of -2 by -3, which gives us 2/3. This visual representation reinforces the concept of equality and how algebraic manipulations maintain the balance of the equation. Remember, the most important thing is to understand the underlying principles and choose the method that resonates with you the most.

Real-World Applications of Solving for x

You might be wondering,