Solving The Equation 3x = 6x - 2 A Comprehensive Guide

In the realm of mathematics, solving for variables is a fundamental skill. Equations are the language of math, and mastering the art of isolating variables allows us to unravel the mysteries they hold. This article delves into the process of solving a linear equation, specifically the equation 3x = 6x - 2. We'll break down the steps, explain the underlying principles, and provide a comprehensive understanding of how to arrive at the solution. Whether you're a student grappling with algebra or simply looking to refresh your math skills, this guide will equip you with the knowledge and confidence to tackle similar problems.

Before we dive into the solution, let's briefly touch upon the nature of linear equations. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. These equations, when graphed, form a straight line, hence the name "linear." The equation 3x = 6x - 2 fits this definition, as it involves the variable 'x' raised to the power of 1. Solving a linear equation means finding the value of the variable that makes the equation true. In simpler terms, we're looking for the number that, when substituted for 'x', will make both sides of the equation equal.

Now, let's embark on the journey of solving the equation 3x = 6x - 2. We'll meticulously walk through each step, providing explanations along the way to ensure clarity.

Step 1: Combine Like Terms

The first crucial step in solving any equation is to consolidate like terms. Like terms are terms that contain the same variable raised to the same power. In our equation, we have two terms involving 'x': 3x and 6x. To combine them, we need to bring them to the same side of the equation. A common strategy is to move the term with the smaller coefficient. In this case, 3x has a smaller coefficient than 6x. To move 3x from the left side to the right side, we subtract 3x from both sides of the equation. This maintains the balance of the equation, a fundamental principle in algebra.

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

Simplifying both sides, we get:

0 = 3x - 2

Step 2: Isolate the Variable Term

Our next goal is to isolate the term containing the variable 'x'. Currently, the 3x term is being subtracted by 2. To isolate 3x, we need to eliminate the -2. We achieve this by performing the inverse operation, which is addition. We add 2 to both sides of the equation:

0 + 2 = 3x - 2 + 2

This simplifies to:

2 = 3x

Now, the variable term 3x is isolated on the right side of the equation.

Step 3: Solve for x

The final step is to solve for 'x' itself. The x is currently being multiplied by 3. To isolate 'x', we perform the inverse operation, which is division. We divide both sides of the equation by 3:

2 / 3 = 3x / 3

This simplifies to:

x = 2/3

Therefore, the solution to the equation 3x = 6x - 2 is x = 2/3.

To ensure the accuracy of our solution, it's always a good practice to verify it. We do this by substituting the value we found for 'x' back into the original equation.

Original Equation:

3x = 6x - 2

Substitute x = 2/3:

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

Simplify:

2 = 4 - 2

2 = 2

Since both sides of the equation are equal, our solution x = 2/3 is correct. This verification step provides confidence in our answer and highlights the importance of checking our work in mathematics.

While the method outlined above is a standard approach, there's often more than one way to solve an equation. Let's explore an alternative method to solve 3x = 6x - 2. In this method, we'll choose to move the 6x term to the left side of the equation instead of the 3x term. This demonstrates the flexibility in algebraic manipulation.

Step 1: Move 6x to the Left Side

To move 6x from the right side to the left side, we subtract 6x from both sides of the equation:

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

Simplifying both sides, we get:

-3x = -2

Step 2: Solve for x

Now, to isolate 'x', we divide both sides of the equation by -3:

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

This simplifies to:

x = 2/3

As you can see, we arrive at the same solution, x = 2/3, using this alternative method. This highlights that different approaches can lead to the same correct answer in mathematics. The choice of method often depends on personal preference and the specific equation's structure.

Solving equations involves careful manipulation, and it's easy to make mistakes if one isn't cautious. Let's identify some common pitfalls to avoid when solving equations like 3x = 6x - 2:

1. Incorrectly Combining Like Terms: A frequent error is adding or subtracting terms that are not like terms. For example, one might mistakenly try to combine 3x with the constant term -2. Remember, only terms with the same variable raised to the same power can be combined.
2. Forgetting to Perform Operations on Both Sides: The golden rule of equation solving is that any operation performed on one side must be performed on the other side to maintain balance. For instance, if you subtract 3x from the left side, you must also subtract it from the right side. Failing to do so will lead to an incorrect solution.
3. Incorrectly Applying the Order of Operations: Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Applying operations in the wrong order can lead to errors.
4. Sign Errors: Pay close attention to the signs of the terms. A common mistake is to drop a negative sign or to apply it incorrectly. For example, when dividing -2 by -3, the result is positive 2/3, not negative.
5. Not Verifying the Solution: As demonstrated earlier, verifying the solution is crucial to ensure accuracy. Substituting the solution back into the original equation can reveal errors that might have been overlooked during the solving process.

By being mindful of these common mistakes, you can significantly improve your accuracy and confidence in solving equations.

Solving for 'x' isn't just an abstract mathematical exercise; it has numerous real-world applications. Understanding how to isolate variables in equations is fundamental to various fields, including:

1. Science: In physics, chemistry, and other sciences, equations are used to model phenomena and relationships. Solving for 'x' might involve finding an unknown force, concentration, or time.
2. Engineering: Engineers use equations extensively in design and analysis. Solving for 'x' might be necessary to determine the required dimensions of a structure, the optimal flow rate in a system, or the electrical current in a circuit.
3. Economics and Finance: Economic models and financial calculations often involve equations. Solving for 'x' could help determine the equilibrium price in a market, the return on an investment, or the loan payment amount.
4. Computer Science: Algorithms and programming rely heavily on mathematical logic. Solving for 'x' might be used in optimization problems, data analysis, or game development.
5. Everyday Life: Even in everyday situations, solving for 'x' can be useful. For example, you might use it to calculate the discount on a sale item, determine the amount of ingredients needed to scale a recipe, or figure out the time it will take to travel a certain distance.

These examples illustrate that the ability to solve for 'x' is a valuable skill that extends far beyond the classroom. It's a tool for problem-solving and critical thinking in a wide range of contexts.

In this article, we've thoroughly explored the process of solving the equation 3x = 6x - 2. We've broken down the steps, explained the underlying principles, and provided an alternative method for solving the equation. We've also highlighted common mistakes to avoid and discussed the wide-ranging applications of solving for 'x'.

Mastering the art of solving equations is a cornerstone of mathematical proficiency. It empowers you to tackle complex problems, make informed decisions, and navigate the quantitative aspects of the world around you. By understanding the concepts and practicing the techniques outlined in this article, you'll be well-equipped to confidently solve for 'x' in a variety of equations and contexts. Remember, mathematics is a journey of discovery, and each equation solved is a step forward on that path.