Solving For Y In 11y - 6y = 35 A Step-by-Step Guide
In the realm of mathematics, solving for variables is a fundamental skill. This article will provide a detailed, step-by-step guide on how to solve the equation 11y - 6y = 35, simplifying the answer as much as possible. We will explore the underlying principles of algebraic manipulation, ensuring a clear understanding of each step involved. Whether you are a student grappling with algebra or simply looking to refresh your mathematical skills, this guide will equip you with the knowledge and confidence to tackle similar problems. Understanding the basics of algebraic equations is the cornerstone of mathematical problem-solving, and this example serves as an excellent starting point. The process involves combining like terms, isolating the variable, and performing arithmetic operations to arrive at the solution. By mastering these techniques, you can unlock the ability to solve a wide array of algebraic challenges. The beauty of algebra lies in its systematic approach, where each step is logically connected to the previous one, leading to a precise and verifiable answer. This article aims to demystify the process, making it accessible and engaging for learners of all levels. So, let's embark on this mathematical journey and unravel the solution to the equation 11y - 6y = 35.
Simplifying the Equation
To begin solving for y in the equation 11y - 6y = 35, the first crucial step is to simplify the equation by combining like terms. In this case, 11y and -6y are like terms because they both contain the variable y. Combining like terms involves performing the indicated operation on their coefficients. In this instance, we subtract 6 from 11, resulting in 5. Therefore, 11y - 6y simplifies to 5y. Now, the equation is reduced to a much simpler form: 5y = 35. This simplification is a fundamental technique in algebra, allowing us to condense complex expressions into more manageable forms. Simplifying equations is not just about making them shorter; it's about revealing the underlying structure and making the problem easier to solve. This step is crucial because it reduces the number of operations needed to isolate the variable, thus minimizing the chances of making errors. By combining like terms, we've transformed the original equation into a much more straightforward one, setting the stage for the next step in solving for y. Understanding this process is essential for tackling more complex algebraic problems in the future. The ability to identify and combine like terms is a cornerstone of algebraic manipulation, enabling us to efficiently solve equations and inequalities.
Isolating the Variable
After simplifying the equation to 5y = 35, the next pivotal step is to isolate the variable y. Isolating a variable means getting it by itself on one side of the equation. In this scenario, y is being multiplied by 5. To undo this multiplication and isolate y, we perform the inverse operation, which is division. We divide both sides of the equation by 5. This is a fundamental principle of algebra: whatever operation you perform on one side of the equation, you must also perform on the other side to maintain the equality. Dividing both sides of 5y = 35 by 5, we get (5y)/5 = 35/5. On the left side, the 5 in the numerator and the 5 in the denominator cancel each other out, leaving us with just y. On the right side, 35 divided by 5 equals 7. Therefore, the equation simplifies to y = 7. Isolating the variable is the key to solving any algebraic equation. It's the process of unraveling the operations that have been applied to the variable, step by step, until the variable stands alone, revealing its value. This step often involves using inverse operations, such as addition to undo subtraction, multiplication to undo division, and vice versa. The goal is to create a situation where the variable is the only term on one side of the equation, and a numerical value is on the other side. This process requires careful attention to detail and a solid understanding of algebraic principles.
The Solution: y = 7
Having isolated the variable y, we have arrived at the solution: y = 7. This means that the value of y that satisfies the original equation 11y - 6y = 35 is 7. To verify this solution, we can substitute 7 back into the original equation in place of y and check if both sides of the equation are equal. Substituting y = 7 into the original equation, we get 11(7) - 6(7) = 35. Simplifying the left side, we have 77 - 42 = 35, which is indeed true. This verification confirms that our solution, y = 7, is correct. Finding the solution is the culmination of the entire problem-solving process. It's the point where we've successfully manipulated the equation to reveal the value of the unknown variable. However, it's always a good practice to verify the solution by substituting it back into the original equation. This step ensures that we haven't made any errors along the way and that our solution is accurate. The solution y = 7 is a single numerical value, indicating that this is a linear equation with a unique solution. In more complex equations, there might be multiple solutions or no solutions at all. However, in this case, we've found a clear and definitive answer.
Importance of Showing Your Work
While arriving at the correct solution is the ultimate goal, it is equally important to show your work in mathematics. Demonstrating each step of the solving process provides clarity and allows for easier identification of any potential errors. When you show your work, you create a transparent pathway from the initial equation to the final solution. This not only helps you understand the logic behind each step but also enables others to follow your reasoning. In an educational setting, showing your work is often a significant part of the grading criteria. It allows teachers to assess your understanding of the concepts and techniques involved, even if you happen to make a minor calculation error. Showing your work is a critical skill in mathematics for several reasons. Firstly, it helps you to organize your thoughts and approach the problem systematically. By writing down each step, you're less likely to make careless mistakes. Secondly, it makes it easier to review your work and identify any errors you might have made. If you simply write down the final answer without showing the steps, it can be challenging to track back and find where you went wrong. Thirdly, showing your work is essential for communication. In mathematics, it's not enough to just get the right answer; you also need to be able to explain how you arrived at that answer. Clear and well-organized work provides evidence of your understanding and allows others to learn from your problem-solving process.
Real-World Applications of Solving for Variables
The ability to solve for variables is not just an academic exercise; it has numerous real-world applications across various fields. From calculating the trajectory of a projectile in physics to determining the optimal budget allocation in finance, the principles of algebra are used extensively. Engineers use algebraic equations to design structures, circuits, and machines. Economists use them to model market behavior and predict economic trends. Computer scientists use them to develop algorithms and software. Even in everyday situations, we often use algebraic thinking without realizing it. For instance, when calculating the amount of ingredients needed to scale a recipe or determining the best deal while shopping, we are essentially solving for variables. Solving for variables is a fundamental skill that empowers us to make informed decisions and solve problems in a wide range of contexts. The real-world applications of algebra are vast and varied, touching almost every aspect of our lives. The ability to translate a real-world problem into an algebraic equation and then solve for the unknown variable is a powerful tool. It allows us to quantify relationships, make predictions, and optimize outcomes. Whether it's calculating the monthly payments on a loan, determining the optimal speed to travel to save fuel, or figuring out the dimensions of a garden, algebraic thinking is an invaluable asset. By mastering the techniques of solving for variables, we gain the ability to analyze complex situations and find effective solutions.
Conclusion
In conclusion, solving for y in the equation 11y - 6y = 35 involves simplifying the equation by combining like terms, isolating the variable y by performing inverse operations, and verifying the solution. The solution, y = 7, can be easily verified by substituting it back into the original equation. Mastering the process of solving for variables is crucial for success in mathematics and various real-world applications. By understanding the fundamental principles of algebraic manipulation and practicing regularly, you can develop the skills and confidence to tackle a wide range of mathematical problems. Remember to always show your work, as this not only helps in identifying potential errors but also demonstrates a clear understanding of the problem-solving process. The ability to solve for variables is a cornerstone of mathematical literacy, empowering us to analyze and solve problems in diverse contexts. From simple equations to complex systems, the principles of algebra provide a powerful framework for understanding and manipulating the world around us. So, embrace the challenge, practice diligently, and unlock the potential of algebraic thinking.
