Solving The Linear Equation 6k + 10.5 = 3k + 12 A Step-by-Step Guide
Solving linear equations is a fundamental skill in algebra, and mastering it opens doors to more advanced mathematical concepts. In this comprehensive guide, we will delve into the process of solving the linear equation 6k + 10.5 = 3k + 12, providing a step-by-step approach that ensures clarity and understanding. This article is designed for students, educators, and anyone looking to refresh their algebra skills. We'll break down each step, explain the underlying principles, and highlight common pitfalls to avoid. By the end of this guide, you'll not only know the solution to this specific equation but also have a solid foundation for tackling other linear equations.
Understanding Linear Equations
Before diving into the solution, let's establish a clear understanding of what linear equations are. 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 are called "linear" because when graphed, they form a straight line. The general form of a linear equation in one variable is ax + b = c, where a, b, and c are constants, and x is the variable. Our equation, 6k + 10.5 = 3k + 12, fits this form, with k being the variable we need to solve for. Understanding this basic structure is crucial because it dictates the methods we use to find the solution. Linear equations are the building blocks of many mathematical and real-world problems, appearing in physics, engineering, economics, and computer science. Solving them involves isolating the variable on one side of the equation, which we achieve by performing the same operations on both sides to maintain balance. The goal is to simplify the equation until we have k equal to a specific value. This value represents the solution to the equation – the number that, when substituted for k, makes the equation true. The ability to solve linear equations is not just a mathematical skill; it's a critical thinking tool that helps in problem-solving across various disciplines. This is because the logical steps involved in isolating a variable – adding, subtracting, multiplying, and dividing – are fundamental to analytical thinking. As we proceed, we will see how these steps are applied in the context of our specific equation, making the process more concrete and easier to grasp. By mastering linear equations, you're not just learning algebra; you're honing your problem-solving abilities, which are essential in both academic and professional settings.
Step 1: Grouping Like Terms
The first step in solving the equation 6k + 10.5 = 3k + 12 is to group the like terms. Like terms are those that contain the same variable or are constants. In our equation, the terms with the variable k are 6k and 3k, and the constant terms are 10.5 and 12. The goal is to bring all terms involving k to one side of the equation and all constant terms to the other side. This is achieved by performing the same operation on both sides of the equation, ensuring that the equation remains balanced. We can start by subtracting 3k from both sides of the equation. This moves the 3k term from the right side to the left side. The equation then becomes: 6k - 3k + 10.5 = 3k - 3k + 12. Simplifying this, we get 3k + 10.5 = 12. Next, we need to move the constant term 10.5 from the left side to the right side. We do this by subtracting 10.5 from both sides of the equation: 3k + 10.5 - 10.5 = 12 - 10.5. This simplifies to 3k = 1.5. By grouping like terms, we have effectively isolated the term with the variable k on one side of the equation and the constant term on the other side. This makes it easier to solve for k in the next step. The principle of maintaining balance in an equation is crucial here. Whatever operation we perform on one side, we must also perform on the other side. This ensures that the equality holds true throughout the process. Grouping like terms is a fundamental technique in algebra, and it's a skill that will be used extensively in solving various types of equations. It's a systematic way to simplify the equation and bring it closer to the solution. By mastering this step, you're laying a strong foundation for more complex algebraic manipulations. The ability to identify and group like terms efficiently is a key component of algebraic fluency, making the process of solving equations smoother and more intuitive.
Step 2: Isolating the Variable
After grouping like terms, our equation stands as 3k = 1.5. The next step is to isolate the variable k. This means getting k by itself on one side of the equation. Currently, k is being multiplied by 3. To isolate k, we need to undo this multiplication. The inverse operation of multiplication is division. Therefore, we will divide both sides of the equation by 3. This maintains the balance of the equation, as we are performing the same operation on both sides. Dividing both sides by 3, we get: (3k) / 3 = 1.5 / 3. On the left side, the 3 in the numerator and the 3 in the denominator cancel each other out, leaving us with just k. On the right side, 1.5 divided by 3 is 0.5. So, the equation simplifies to: k = 0.5. We have now successfully isolated the variable k and found its value. This is the solution to the equation 6k + 10.5 = 3k + 12. Isolating the variable is a critical step in solving any algebraic equation. It involves using inverse operations to undo the operations that are being applied to the variable. This step requires a clear understanding of the relationship between different mathematical operations. Multiplication and division are inverse operations, as are addition and subtraction. The choice of which operation to use depends on what is currently being done to the variable. In our case, since k was being multiplied by 3, we used division to isolate it. This principle of using inverse operations is a cornerstone of algebraic problem-solving and is applicable to a wide range of equations, not just linear ones. Mastering this step allows you to systematically unravel equations and reveal the value of the unknown variable. The solution k = 0.5 means that if we substitute 0.5 for k in the original equation, both sides of the equation will be equal. This is a way to check our solution and ensure that we have not made any errors in our calculations. The process of isolating the variable is not just about finding a number; it's about understanding the underlying structure of the equation and how different operations interact with each other.
Step 3: Verifying the Solution
To ensure the accuracy of our solution, it's always a good practice to verify it. We found that k = 0.5 is the solution to the equation 6k + 10.5 = 3k + 12. To verify this, we substitute 0.5 for k in the original equation and check if both sides are equal. Substituting k = 0.5 into the left side of the equation, we get: 6(0.5) + 10.5. This simplifies to 3 + 10.5, which equals 13.5. Now, we substitute k = 0.5 into the right side of the equation: 3(0.5) + 12. This simplifies to 1.5 + 12, which also equals 13.5. Since both sides of the equation are equal when k = 0.5, our solution is correct. This verification step is crucial because it catches any potential errors that may have occurred during the solving process. It provides confidence in the accuracy of the solution and reinforces the understanding of the equation. Verification is not just a formality; it's an integral part of the problem-solving process. It demonstrates a thorough understanding of the equation and the steps taken to solve it. By verifying our solution, we are essentially proving that the value we found for k satisfies the original equation. This is a fundamental concept in algebra – the solution to an equation is the value that makes the equation true. The verification process also helps in developing a deeper understanding of the equation's structure and the relationship between its terms. It's a way to connect the abstract concept of solving an equation to the concrete reality of substituting values and checking for equality. This connection is essential for building a strong foundation in algebra and for applying these skills to more complex problems. In addition to catching errors, verification can also help in identifying alternative methods for solving the equation. By substituting the solution back into the original equation, we may gain insights into other ways the equation could have been simplified or manipulated. This can lead to a more flexible and adaptable approach to problem-solving in the future.
Conclusion
In this guide, we have successfully solved the linear equation 6k + 10.5 = 3k + 12. We followed a step-by-step approach, starting with grouping like terms, then isolating the variable, and finally verifying the solution. Our solution, k = 0.5, is the correct answer. This process demonstrates the fundamental principles of solving linear equations and highlights the importance of each step. By understanding these principles, you can confidently tackle a wide range of linear equations. Solving linear equations is a foundational skill in mathematics, and mastering it is crucial for success in more advanced topics. The steps we have outlined – grouping like terms, isolating the variable, and verifying the solution – are applicable to virtually any linear equation. The key is to approach each equation systematically and pay close attention to the operations being performed. Algebra is not just about memorizing formulas; it's about understanding the underlying logic and applying it to solve problems. The ability to solve linear equations is not just valuable in mathematics; it's also a valuable life skill. It teaches logical thinking, problem-solving, and attention to detail – qualities that are essential in many areas of life. Whether you're balancing a budget, planning a project, or making important decisions, the skills you learn in algebra can help you think critically and make informed choices. We encourage you to practice solving linear equations regularly to solidify your understanding and build your confidence. The more you practice, the more comfortable you will become with the process, and the easier it will be to solve even more complex equations. Remember, mathematics is a journey, and each equation you solve is a step forward. The principles and techniques discussed here form a solid foundation for your continued mathematical growth. By mastering linear equations, you're not just learning a mathematical skill; you're developing a mindset of logical problem-solving that will serve you well in all your future endeavors. Keep practicing, keep exploring, and keep challenging yourself – the world of mathematics is vast and rewarding, and you're well on your way to making significant strides in it.
Final Answer: The final answer is (A)
