Solving For X In The Equation -6x + 4x = 8

In the realm of algebraic equations, finding the value of an unknown variable is a fundamental skill. This article delves into the process of solving the equation -6x + 4x = 8, providing a step-by-step guide to arrive at the correct solution. Understanding how to solve such equations is crucial for various mathematical and scientific applications, laying the foundation for more complex problem-solving scenarios.

To begin, let's address the core question: What is the value of x in the equation -6x + 4x = 8? This seemingly simple equation holds the key to understanding basic algebraic principles. By systematically applying the rules of algebra, we can isolate the variable 'x' and determine its numerical value. The process involves combining like terms, simplifying the equation, and performing operations on both sides to maintain equality. It's like solving a puzzle, where each step brings us closer to the final answer. This exploration isn't just about finding the solution; it's about understanding the underlying logic and techniques that empower us to tackle a wide range of mathematical problems. So, let's embark on this journey of algebraic discovery and unveil the value of 'x' in this equation. The significance of mastering these skills extends far beyond the classroom, playing a pivotal role in fields like engineering, finance, and computer science. By understanding the mechanics of equation-solving, we equip ourselves with a powerful tool for analytical thinking and problem-solving in the real world. In the following sections, we'll break down the equation step by step, ensuring a clear and comprehensive understanding of the solution process. From combining like terms to isolating the variable, each step will be explained with precision, making it accessible to learners of all levels.

Simplifying the Equation

The initial step in solving the equation -6x + 4x = 8 involves simplifying the left-hand side. We achieve this by combining the 'x' terms. Think of it as grouping similar objects together – in this case, we're grouping the 'x' variables. This process is a cornerstone of algebraic manipulation, allowing us to reduce complex expressions into simpler, more manageable forms. The ability to simplify expressions is not just a mathematical skill; it's a valuable tool for organizing and streamlining information in various contexts. Whether it's balancing a budget or planning a project, the principle of simplifying and organizing elements is universally applicable. In this specific equation, we have two terms involving 'x': -6x and +4x. These are like terms because they both contain the same variable, 'x', raised to the same power (which is 1 in this case). To combine them, we simply add their coefficients, which are the numbers multiplying the 'x'. So, we add -6 and 4. Understanding the rules of integer addition is crucial here. Adding a negative number is the same as subtracting its positive counterpart. Therefore, -6 + 4 is the same as 4 - 6, which equals -2. This seemingly small step is a significant milestone in our journey to solve for 'x'. By combining these terms, we've effectively reduced the complexity of the equation, making it easier to isolate the variable. The simplified form of the left-hand side is -2x. Now, our equation looks much cleaner: -2x = 8. This simplified equation retains the same mathematical meaning as the original but is easier to work with. It's like decluttering a room – by removing unnecessary elements, we can focus on the essentials. In mathematics, simplification is the key to clarity and efficiency. It allows us to see the underlying structure of an equation and identify the most effective path to the solution. The next step will involve isolating 'x' completely, but this simplification has set the stage for that process.

Isolating the Variable x

After simplifying the equation to -2x = 8, the next crucial step is isolating the variable 'x'. This means getting 'x' by itself on one side of the equation. To achieve this, we need to undo the operation that is currently being applied to 'x'. In this case, 'x' is being multiplied by -2. The inverse operation of multiplication is division. Therefore, to isolate 'x', we will divide both sides of the equation by -2. This is a fundamental principle of algebra: whatever operation we perform on one side of the equation, we must perform the same operation on the other side to maintain the equality. This ensures that the equation remains balanced, like a scale where both sides must weigh the same. If we were to only divide one side, the equation would become unbalanced and the solution would be incorrect. Dividing both sides by the same number is like distributing a shared resource equally – everyone gets their fair share. So, we divide both -2x and 8 by -2. On the left-hand side, -2x divided by -2 simplifies to 'x'. This is because a number divided by itself is 1, and 1 multiplied by 'x' is simply 'x'. This is the essence of isolating the variable – we've effectively removed the coefficient -2 from 'x', leaving it all alone on the left side. On the right-hand side, we have 8 divided by -2. When dividing a positive number by a negative number, the result is negative. 8 divided by 2 is 4, so 8 divided by -2 is -4. This step highlights the importance of understanding the rules of sign when performing arithmetic operations. A simple mistake with signs can lead to an incorrect solution. Now, we have successfully isolated 'x' and found its value. The equation has been transformed into x = -4. This is our solution – the value of 'x' that makes the original equation true. It's like finding the missing piece of a puzzle – we've uncovered the value that fits perfectly.

Verifying the Solution

Once we've found a potential solution, it's essential to verify its correctness. In this case, we've determined that x = -4 is the solution to the equation -6x + 4x = 8. To verify this, we substitute -4 for 'x' in the original equation and check if both sides of the equation are equal. This process is like checking our work – it ensures that we haven't made any mistakes along the way. Verification is a critical step in problem-solving, not just in mathematics but in various fields. It's about ensuring accuracy and building confidence in our results. Imagine building a bridge – you wouldn't want to skip the verification process, as it could have disastrous consequences. Similarly, in mathematics, verifying our solutions helps us avoid errors and develop a deeper understanding of the concepts involved. So, let's substitute -4 for 'x' in the original equation: -6x + 4x = 8. Replacing 'x' with -4, we get: -6(-4) + 4(-4) = 8. Now, we perform the multiplications. Remember that multiplying two negative numbers results in a positive number. -6 multiplied by -4 is 24. Similarly, multiplying a positive number by a negative number results in a negative number. 4 multiplied by -4 is -16. So, our equation now looks like this: 24 + (-16) = 8. Adding a negative number is the same as subtracting its positive counterpart, so 24 + (-16) is the same as 24 - 16. 24 - 16 equals 8. Therefore, the left-hand side of the equation simplifies to 8. Now, we compare this to the right-hand side of the equation, which is also 8. Since both sides are equal, our solution x = -4 is verified. This confirms that our step-by-step process was correct and that we have indeed found the value of 'x' that satisfies the equation. The verification process not only confirms the solution but also reinforces our understanding of the equation and the algebraic principles involved. It's a valuable learning experience that builds confidence and competence in problem-solving.

Conclusion: The Value of x is -4

In conclusion, by systematically simplifying the equation -6x + 4x = 8, isolating the variable 'x', and verifying our solution, we have determined that the value of x is -4. This journey through algebraic manipulation highlights the importance of understanding fundamental principles such as combining like terms, performing inverse operations, and maintaining equality. These skills are not only essential for solving equations but also for tackling a wide range of mathematical and scientific problems. The process of solving for 'x' is a microcosm of the broader problem-solving process – it involves breaking down a complex problem into smaller, more manageable steps, applying logical reasoning, and verifying the results. This approach is applicable in various aspects of life, from making informed decisions to designing innovative solutions. The ability to manipulate equations and solve for unknown variables is a powerful tool for analytical thinking and critical reasoning. It empowers us to understand relationships between quantities, make predictions, and solve real-world problems. Whether it's calculating the trajectory of a projectile or determining the optimal investment strategy, the principles of algebra are at the heart of many important applications. Moreover, the process of verification underscores the importance of accuracy and attention to detail. In mathematics, a small mistake can lead to a completely incorrect solution. By verifying our work, we ensure that our results are reliable and that our understanding is sound. This habit of verification is valuable in any field, from scientific research to financial analysis. It's about building trust in our own work and ensuring the integrity of our findings. So, the next time you encounter an algebraic equation, remember the steps we've outlined – simplify, isolate, and verify. With practice and persistence, you'll become proficient in solving for 'x' and unlocking the power of algebra.

Answer: A. -4