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

In the realm of mathematics, solving for 'x' in a given equation is a fundamental skill. It's the cornerstone of algebra and plays a crucial role in various scientific and engineering disciplines. Our focus here is to dissect the equation 3x = 6x - 2, providing a comprehensive, step-by-step solution and shedding light on the underlying principles. This exploration will not only aid in solving this specific equation but will also equip you with the tools to tackle similar algebraic challenges. This process requires us to isolate the variable 'x' on one side of the equation. To achieve this, we'll employ algebraic manipulations, ensuring that the equation remains balanced at each stage. Understanding the mechanics of these manipulations is key to mastering algebra. In essence, we aim to simplify the equation, bringing all terms involving 'x' to one side and all constant terms to the other, thereby revealing the value of 'x'. This journey through the equation will be methodical, explaining each step in detail, thus building a solid understanding of how to approach and solve linear equations. The beauty of algebra lies in its logical structure, where each operation is a deliberate move towards simplification and solution. By following these steps, you'll gain confidence in your ability to solve equations, a skill that transcends the classroom and finds application in real-world problem-solving.

Step-by-Step Solution

1. Understanding the Equation

The equation we're tackling is 3x = 6x - 2. This is a linear equation, a type that forms a straight line when graphed. The objective is to find the value of 'x' that makes the equation true. In other words, we seek the value of 'x' that, when substituted back into the equation, will result in both sides being equal. Linear equations are fundamental in algebra, serving as building blocks for more complex mathematical concepts. They appear in various contexts, from calculating distances and speeds to determining financial investments and economic models. The simplicity of linear equations belies their power and ubiquity in mathematical and practical applications. Understanding how to solve them is, therefore, an essential skill for anyone venturing into mathematics, science, or engineering. This initial step of understanding the equation sets the stage for the subsequent steps, ensuring we approach the problem with a clear goal in mind: to isolate 'x' and uncover its value.

2. Isolate the x Terms

The initial move in solving for 'x' is to gather all terms containing 'x' on one side of the equation. A strategic approach is to subtract 6x from both sides of the equation. This action eliminates the 'x' term from the right side, moving us closer to isolating 'x' on one side. By subtracting 6x from both sides, we maintain the equation's balance, a crucial principle in algebra. What we do to one side, we must do to the other to preserve the equality. This step demonstrates the fundamental algebraic principle of maintaining balance in equations. The result of this subtraction is a new equation: 3x - 6x = -2. This simplified equation brings us one step closer to our goal. It consolidates the 'x' terms, making the next steps more manageable. This process of isolating variables is a core technique in algebra, applicable to a wide range of equation-solving scenarios. By mastering this technique, you enhance your ability to manipulate equations effectively and efficiently.

3. Simplify the Equation

Following the isolation of 'x' terms, the next crucial step is to simplify the equation. This involves combining like terms, specifically the 'x' terms on the left side of the equation. In our case, we have 3x - 6x. Performing this subtraction, we arrive at -3x. The equation now reads -3x = -2. This simplification is a pivotal moment in the solution process. By combining like terms, we reduce the equation to its most basic form, making it easier to isolate 'x' completely. Simplification is not just about reducing terms; it's about clarifying the relationship between the variable and the constants in the equation. This step highlights the importance of arithmetic skills in algebra. The ability to add, subtract, multiply, and divide correctly is essential for accurate equation solving. The simplified equation presents a clearer path to the final solution, where 'x' will stand alone, revealing its value.

4. Solve for x

With the simplified equation -3x = -2, the final step in solving for 'x' is to isolate 'x' entirely. To achieve this, we divide both sides of the equation by the coefficient of 'x', which is -3. This division is a critical step, as it undoes the multiplication of 'x' by -3, effectively freeing 'x'. Dividing both sides by -3 ensures the equation remains balanced, adhering to the fundamental principles of algebra. The result of this division is x = -2 / -3. A negative number divided by a negative number yields a positive result, thus simplifying the fraction to x = 2/3. This is the solution to the equation. We have successfully isolated 'x' and determined its value. This final step underscores the inverse relationship between multiplication and division in algebra. By understanding and applying this relationship, we can efficiently solve for variables in equations. The solution x = 2/3 represents the value that, when substituted back into the original equation, will make the equation true. This verification is a crucial step in the problem-solving process, ensuring the accuracy of the solution.

Final Answer

Therefore, the solution to the equation 3x = 6x - 2 is x = 2/3. This result represents the value of 'x' that satisfies the equation, making both sides equal. In the context of linear equations, this solution is a specific point on the number line that, when substituted into the equation, confirms its validity. The journey to find this solution involved several key steps: understanding the equation, isolating 'x' terms, simplifying the equation, and finally, solving for 'x'. Each step is a building block in the process, highlighting the logical and methodical approach required in algebra. The solution x = 2/3 is not just a number; it's the culmination of these algebraic manipulations, a testament to the power of equation solving. It demonstrates the precision and elegance of mathematics, where a series of logical steps leads to a definitive answer. This final answer reinforces the importance of understanding and applying algebraic principles to solve mathematical problems effectively.

Conclusion

In conclusion, solving the equation 3x = 6x - 2 to find x = 2/3 showcases the fundamental principles of algebra. The process, involving isolation of variables, simplification, and maintaining equation balance, is a cornerstone of mathematical problem-solving. This exercise not only provides a concrete answer but also reinforces the systematic approach necessary for tackling algebraic challenges. The steps taken – from rearranging terms to performing arithmetic operations – highlight the interconnectedness of mathematical concepts. Each operation is deliberate, contributing to the overall goal of isolating and determining the value of 'x'. Mastering these techniques is crucial for anyone venturing into more advanced mathematics, science, or engineering fields. The ability to manipulate equations with confidence and precision is a skill that transcends the classroom, finding application in various real-world scenarios. This solution exemplifies the power of algebra as a tool for uncovering hidden values and relationships within mathematical expressions. The process of solving for 'x' is a microcosm of mathematical inquiry, where logic, precision, and perseverance lead to a definitive solution.