Solving For S In The Equation 3*(18-97) + 540/80 + 7s = 290
In the realm of mathematics, solving equations is a fundamental skill. Whether you're dealing with simple linear equations or more complex expressions, the ability to isolate variables and determine their values is crucial. In this comprehensive guide, we'll dissect the process of solving for the variable 's' in a given equation. Our focus will be on clarity, step-by-step instructions, and ensuring you grasp the underlying principles. The equation we will tackle is: 3*(18-97) + 540/80 + 7s = 290. We'll break down each step, making it easy to follow and understand. We'll explore the order of operations, simplification techniques, and how to effectively isolate the variable 's' to find its value. We'll also delve into the common pitfalls and mistakes to avoid when solving equations. The goal is not just to provide you with the answer but to equip you with the knowledge and confidence to solve similar problems independently. We'll also discuss the importance of checking your work to ensure accuracy and highlight how these skills apply to various mathematical and real-world scenarios. By the end of this guide, you will be able to confidently tackle equations of this nature and have a solid foundation for more advanced mathematical concepts. So, let's embark on this mathematical journey together and unravel the mystery behind solving for 's'. This is more than just an exercise in algebra; it's a journey into the heart of problem-solving, where every step we take builds our understanding and sharpens our mathematical acumen. Prepare to transform from a solver of equations into a master of mathematical thinking.
Breaking Down the Equation: A Step-by-Step Approach
When faced with an equation like 3*(18-97) + 540/80 + 7s = 290, the key to success lies in a systematic approach. First, we must understand the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which we perform calculations to ensure a consistent and accurate result. Let's begin by simplifying the expression within the parentheses: (18-97). This subtraction yields -79, so our equation now looks like this: 3*(-79) + 540/80 + 7s = 290. Next, we address the multiplication and division operations. Multiplying 3 by -79 gives us -237, and dividing 540 by 80 results in 6.75. Our equation is now simplified to: -237 + 6.75 + 7s = 290. This step is crucial because it reduces the complexity of the equation, making it easier to manage. Now, we combine the constant terms on the left side of the equation. Adding -237 and 6.75 gives us -230.25. So, the equation becomes: -230.25 + 7s = 290. We are one step closer to isolating 's'. Remember, each step we take is a deliberate move towards simplifying the equation and bringing 's' closer to its solved value. This methodical approach not only helps us avoid errors but also deepens our understanding of the equation's structure. In the next section, we will focus on isolating 's' completely, revealing its value and completing our mathematical quest.
Isolating the Variable: Unveiling the Value of 's'
Now that we've simplified the equation to -230.25 + 7s = 290, the next crucial step is to isolate the variable 's'. Isolating a variable means getting it by itself on one side of the equation. To do this, we need to undo the operations that are being performed on 's'. In this case, 's' is being multiplied by 7 and has -230.25 added to it. The first step is to eliminate the addition of -230.25. We do this by adding 230.25 to both sides of the equation. This maintains the balance of the equation, a fundamental principle in algebra. Adding 230.25 to both sides, we get: -230.25 + 230.25 + 7s = 290 + 230.25. This simplifies to 7s = 520.25. Notice how adding the same value to both sides effectively cancels out the -230.25 on the left side, bringing us closer to isolating 's'. The next step is to undo the multiplication by 7. We achieve this by dividing both sides of the equation by 7. This gives us: (7s)/7 = 520.25/7. Performing this division, we find that s = 74.32 (approximately). Therefore, the value of 's' that satisfies the equation is approximately 74.32. This process of isolating the variable is a cornerstone of solving equations. It's a methodical approach that involves carefully undoing operations while maintaining the equation's balance. In the next section, we will verify our solution, ensuring its accuracy and solidifying our understanding of the process. This is a critical step in any mathematical problem-solving endeavor.
Verification: Ensuring Accuracy in Our Solution
After solving for a variable, the verification step is paramount to ensure the accuracy of our solution. To verify our solution for 's', which we found to be approximately 74.32, we substitute this value back into the original equation: 3*(18-97) + 540/80 + 7s = 290. Substituting s = 74.32, we get: 3*(18-97) + 540/80 + 7*(74.32) = 290. Now, we follow the order of operations to evaluate the left side of the equation. First, we solve the parentheses: 18 - 97 = -79. So, the equation becomes: 3*(-79) + 540/80 + 7*(74.32) = 290. Next, we perform the multiplications and division: 3*(-79) = -237 540/80 = 6.75 7*(74.32) = 520.24. The equation now looks like this: -237 + 6.75 + 520.24 = 290. Finally, we perform the additions: -237 + 6.75 + 520.24 = 289.99. The result, 289.99, is very close to 290. The slight discrepancy is due to rounding the value of 's' to 74.32. If we had used a more precise value, we would have obtained an even closer result. This verification process confirms that our solution for 's' is indeed accurate. It demonstrates the importance of not just finding a solution but also ensuring its correctness. By substituting the solved value back into the original equation, we can catch any errors made during the solving process. In the next section, we will discuss the significance of this solution and its implications in broader mathematical contexts.
Significance and Applications: The Power of Solving Equations
Understanding how to solve equations like 3*(18-97) + 540/80 + 7s = 290 extends far beyond the classroom. The ability to solve equations is a fundamental skill that underpins many areas of mathematics, science, engineering, and even everyday life. In mathematics, equation solving is crucial for algebra, calculus, and other advanced topics. It forms the basis for understanding relationships between variables and predicting outcomes. For example, in physics, equations are used to describe the motion of objects, the behavior of electric circuits, and the properties of light and sound. Solving these equations allows scientists to make predictions and understand the world around us. In engineering, equations are used to design structures, machines, and systems. Engineers use mathematical models to analyze the behavior of these systems and ensure they function correctly. Whether it's calculating the load-bearing capacity of a bridge or designing an efficient engine, equation solving is at the heart of the process. Even in everyday life, we encounter situations where solving equations is helpful. For instance, when budgeting finances, planning a trip, or cooking a recipe, we often need to solve for unknown quantities. Understanding the principles of equation solving empowers us to make informed decisions and solve practical problems. The specific equation we solved in this guide, while seemingly abstract, illustrates the general principles involved in equation solving. The steps we followed – simplifying expressions, isolating variables, and verifying solutions – are applicable to a wide range of equations. By mastering these techniques, you gain a powerful tool for problem-solving in various contexts. In the concluding section, we will summarize the key takeaways from this guide and highlight the importance of continuous learning in mathematics.
Conclusion: Mastering Equations and Embracing Mathematical Growth
In conclusion, our journey through solving the equation 3*(18-97) + 540/80 + 7s = 290 has highlighted the importance of a systematic approach in mathematics. We've seen how breaking down a complex problem into smaller, manageable steps can lead to a clear and accurate solution. We started by understanding the order of operations (PEMDAS), which guided us in simplifying the equation. We then focused on isolating the variable 's', a crucial step in solving for its value. By adding and dividing both sides of the equation, we successfully got 's' by itself, revealing its value to be approximately 74.32. The verification step was equally important. By substituting our solution back into the original equation, we confirmed its accuracy and gained confidence in our answer. This process underscores the significance of not just finding a solution but also ensuring its correctness. Beyond the specific equation, we've explored the broader significance of equation solving. We've seen how this skill is fundamental to various fields, including mathematics, science, engineering, and everyday life. The ability to solve equations empowers us to understand relationships between variables, make predictions, and solve practical problems. As you continue your mathematical journey, remember that practice and persistence are key. The more you engage with equations and problem-solving, the more proficient you will become. Embrace challenges as opportunities for growth, and never hesitate to seek help or clarification when needed. Mathematics is a vast and fascinating subject, and mastering equation solving is a significant step towards unlocking its potential. Keep exploring, keep learning, and keep growing your mathematical skills. The world of mathematics is full of wonders, waiting to be discovered.
