Solving The Equation 7h - 5(3h - 8) = -72 Finding The Hours To Dig A Well

Introduction: Unearthing the Solution to a Mathematical Well

In the realm of mathematics, equations serve as powerful tools for modeling real-world scenarios. This article delves into the equation 7h - 5(3h - 8) = -72, which represents the number of hours it takes to dig a well to a level of 72 feet below sea level. This problem allows us to explore the fascinating intersection of algebra and practical problem-solving. Our goal is to meticulously dissect this equation, unravel its complexities, and ultimately determine the value of 'h', which signifies the number of hours required to dig the well. This equation might seem daunting at first glance, but with a systematic approach, we can break it down into manageable steps and arrive at the solution. Understanding the context of the equation is crucial. We're not just manipulating numbers; we're modeling a real-world scenario. The negative sign in -72 indicates a depth below sea level, adding a practical dimension to the mathematical problem. As we proceed, we'll emphasize the importance of each step, ensuring that the underlying logic is clear and easily understood. By the end of this exploration, you will not only know the answer but also grasp the fundamental principles of algebraic problem-solving. This skill is invaluable, extending far beyond the confines of this particular problem and applicable to a wide range of mathematical challenges. The beauty of mathematics lies in its ability to translate real-world situations into abstract equations, and this problem serves as a perfect example of that. Let's embark on this mathematical journey together, unraveling the mystery of the equation and discovering the hours it takes to dig the well.

Dissecting the Equation: A Step-by-Step Approach

To begin, let's revisit the equation: 7h - 5(3h - 8) = -72. The first step in solving this equation involves simplifying the expression on the left-hand side. This requires applying the distributive property, a fundamental concept in algebra. The distributive property states that a(b + c) = ab + ac. In our equation, we need to distribute the -5 across the terms inside the parentheses (3h - 8). This means multiplying -5 by both 3h and -8. Performing this operation, we get -5 * 3h = -15h and -5 * -8 = 40. Remember that multiplying two negative numbers results in a positive number, which is why -5 multiplied by -8 equals 40. Now, let's rewrite the equation with the distributed terms: 7h - 15h + 40 = -72. The next step is to combine like terms. In this equation, 7h and -15h are like terms because they both contain the variable 'h'. Combining them, we have 7h - 15h = -8h. So, our equation now becomes -8h + 40 = -72. We've successfully simplified the left-hand side of the equation, making it easier to isolate the variable 'h'. The importance of meticulously following each step cannot be overstated. A small error in distribution or combining like terms can lead to an incorrect solution. It's always wise to double-check your work at each stage to ensure accuracy. With the equation now in a more manageable form, we're one step closer to finding the value of 'h'. The process of simplification is not just about reducing the complexity of the equation; it's about revealing the underlying structure and making the solution more accessible. Each step we've taken has brought us closer to isolating 'h' and answering the question of how many hours it takes to dig the well. The beauty of algebra lies in this systematic approach, where complex problems are broken down into smaller, solvable steps.

Isolating the Variable: Unveiling the Value of 'h'

Now that we have the simplified equation -8h + 40 = -72, our next goal is to isolate the variable 'h'. This means getting 'h' by itself on one side of the equation. To do this, we need to undo the operations that are affecting 'h'. Currently, 'h' is being multiplied by -8 and has 40 added to it. We'll address the addition first. To undo the addition of 40, we subtract 40 from both sides of the equation. This is a crucial step, as maintaining the balance of the equation is paramount. Whatever operation we perform on one side, we must perform on the other. Subtracting 40 from both sides, we get: -8h + 40 - 40 = -72 - 40. This simplifies to -8h = -112. Notice how subtracting 40 from both sides effectively cancels out the +40 on the left-hand side, leaving us with just the term containing 'h'. Now, 'h' is being multiplied by -8. To undo this multiplication, we divide both sides of the equation by -8. Again, it's essential to perform the same operation on both sides to maintain balance. Dividing both sides by -8, we get: -8h / -8 = -112 / -8. This simplifies to h = 14. Remember that dividing a negative number by a negative number results in a positive number. So, -112 divided by -8 equals 14. We have now successfully isolated the variable 'h' and found its value. The process of isolating a variable is a fundamental skill in algebra. It's like peeling away layers to reveal the core value. Each step we've taken – subtracting 40 and then dividing by -8 – was designed to systematically remove the operations affecting 'h' until it stood alone. The value we've found, h = 14, represents the solution to the equation. But what does this solution mean in the context of our problem? Let's interpret this result in the next section.

Interpreting the Solution: Hours to Dig the Well

We have arrived at the solution h = 14. Now, let's translate this mathematical result back into the real-world context of our problem. Recall that 'h' represents the number of hours it takes to dig the well to a level of 72 feet below sea level. Therefore, our solution, h = 14, means that it will take 14 hours to dig the well. This is a crucial step in the problem-solving process: interpreting the mathematical answer in the context of the original question. It's not enough to simply find the value of 'h'; we must understand what that value signifies in the real world. In this case, 14 hours is a tangible, meaningful answer. It tells us the duration of time required to complete the task of digging the well. This highlights the power of mathematics to model real-world situations and provide concrete answers to practical questions. The equation 7h - 5(3h - 8) = -72 was not just an abstract set of symbols; it was a representation of a real-world scenario. By solving the equation, we've gained insight into the time it takes to dig a well, a piece of information that could be valuable in various contexts. Think about the implications of this. If someone were planning to dig a well, this equation (or a similar one) could help them estimate the time required. This is just one example of how mathematical models can be used to make predictions and inform decisions. Moreover, the process of solving this equation has reinforced important algebraic skills. We've practiced distributing, combining like terms, and isolating variables – all fundamental techniques that are applicable to a wide range of mathematical problems. So, not only have we found the answer to this specific question, but we've also strengthened our problem-solving abilities in general. The interpretation of the solution is the final piece of the puzzle, connecting the abstract world of mathematics to the concrete world around us.

Verification: Ensuring Accuracy of the Solution

To ensure the accuracy of our solution, h = 14, it's crucial to verify it by substituting this value back into the original equation: 7h - 5(3h - 8) = -72. This process, known as verification, is a fundamental step in problem-solving. It provides a check to see if our solution truly satisfies the equation. Let's substitute h = 14 into the equation: 7(14) - 5(3(14) - 8) = -72. First, we perform the operations inside the parentheses: 3(14) = 42, so the expression becomes 7(14) - 5(42 - 8) = -72. Next, we simplify the expression inside the parentheses: 42 - 8 = 34, giving us 7(14) - 5(34) = -72. Now, we perform the multiplications: 7(14) = 98 and 5(34) = 170, so the equation becomes 98 - 170 = -72. Finally, we perform the subtraction: 98 - 170 = -72. This confirms that our solution, h = 14, satisfies the original equation. The left-hand side of the equation equals the right-hand side, demonstrating that our answer is correct. Verification is not just a formality; it's a vital safeguard against errors. It's easy to make a mistake in the process of solving an equation, and verification allows us to catch those mistakes before they lead to incorrect conclusions. It's a testament to the importance of precision and attention to detail in mathematics. By verifying our solution, we can have confidence in our answer and in the process we used to obtain it. It reinforces the idea that mathematics is a logical and consistent system, where solutions can be rigorously checked and confirmed. Moreover, the act of verification deepens our understanding of the equation itself. It forces us to revisit the steps we took and to see how the solution fits into the overall structure of the problem. This can lead to a more intuitive grasp of the mathematical concepts involved. In essence, verification is a powerful tool that enhances both our accuracy and our understanding in problem-solving.

Conclusion: The Power of Algebraic Problem-Solving

In conclusion, we have successfully solved the equation 7h - 5(3h - 8) = -72 and determined that it takes 14 hours to dig the well to a level of 72 feet below sea level. This journey through algebraic problem-solving has highlighted several key concepts and skills. We began by dissecting the equation, carefully applying the distributive property and combining like terms to simplify the expression. We then moved on to isolating the variable 'h', systematically undoing the operations that were affecting it. This involved subtracting from both sides of the equation and dividing both sides, always maintaining balance. Once we found the solution, h = 14, we interpreted it in the context of the problem, understanding that it represented the number of hours required to dig the well. Finally, we verified our solution by substituting it back into the original equation, ensuring its accuracy. This entire process demonstrates the power and elegance of algebraic problem-solving. Equations are not just abstract symbols; they are tools for modeling real-world situations and finding concrete answers. The ability to manipulate equations, isolate variables, and interpret solutions is a valuable skill that extends far beyond the classroom. It's a skill that can be applied in various fields, from engineering and finance to science and technology. Moreover, the problem-solving techniques we've employed here – breaking down complex problems into smaller steps, paying attention to detail, and verifying solutions – are applicable to a wide range of challenges, both mathematical and non-mathematical. This journey has not only given us the answer to a specific question but has also strengthened our problem-solving abilities in general. It has reinforced the importance of precision, logic, and critical thinking. As we move forward, we can apply these skills to tackle new and challenging problems, confident in our ability to dissect, solve, and interpret. The world is full of problems waiting to be solved, and mathematics provides us with the tools to address them effectively. This exploration of the equation 7h - 5(3h - 8) = -72 is just one example of the power of mathematics to illuminate the world around us.