Solving Equations: Well Digging And Mathematical Truths

Hey math enthusiasts! Let's dive into a problem that combines the practical world of digging a well with the abstract world of algebra. We're going to unravel the truth behind the equation $7h - 5(3h - 8) = -72$. This equation represents the time it takes to dig a well to a depth of 72 feet below sea level. Now, let's explore what the solution to this equation really means. The question we're tackling is: Which statement is true about the solution to the equation? This means, we are looking for the correct interpretation of the result we get after solving for h.

Understanding the Equation and the Context

First off, let's break down the given equation: $7h - 5(3h - 8) = -72$. What does it represent? Well, h here stands for the number of hours it takes to dig the well. The equation itself is a mathematical model that describes some aspect of the well-digging process. The different terms likely represent the rate of digging, any delays, or other factors affecting the progress. The result, -72, signifies the final depth of the well, which is 72 feet below sea level. So, when we solve for h, we are essentially finding out how many hours it takes to reach that depth. Pretty cool, right? The context is very important in this case since we're not just solving a random equation; we're trying to figure out a real-world scenario. The real world doesn't always deal with nice, neat whole numbers; sometimes, things come out in fractions or decimals. Consider the rate at which the well is being dug, the equipment used, and the geological conditions. All these variables can impact the rate of digging and influence whether the solution is a whole number, a fraction, or even a decimal. This understanding is crucial when we get to the answer choices because it helps us to interpret the results and choose the correct answer. The process we follow here is pretty standard: simplify, solve for the variable, and interpret the result.

Now, let's go ahead and solve the equation step by step, which is an important process. The original equation is $7h - 5(3h - 8) = -72$. First, we need to apply the distributive property to get rid of the parentheses. So we multiply -5 by both terms inside the parentheses. That gives us $7h - 15h + 40 = -72$. Next, we combine like terms on the left side of the equation. 7h - 15h is -8h. Thus, the equation simplifies to $-8h + 40 = -72$. Now, to isolate the variable term, we subtract 40 from both sides. This gives us $-8h = -112$. Finally, to solve for h, we divide both sides by -8. This means $h = 14$. The solution to the equation is h = 14. Now that we have the result, we know that it takes 14 hours to dig the well to a depth of 72 feet below sea level. The math part is done. The main thing is to now consider the practical implications of this number and to decide on the appropriate answer. Keep in mind that the number of hours represents a duration, and in this context, it is a perfectly reasonable answer. So, the right answer should reflect that.

Evaluating the Answer Choices

Now, we need to evaluate some statements. Let's analyze the possible answers. The original prompt asks us, “Which statement is true about the solution to the equation?” Based on the problem's context, any of the choices may be right or wrong. Here's how we can analyze each one, with the result h = 14 in mind:

If the options contain the statement that it cannot be a fraction or decimal because the depth of the well is negative, it's incorrect. Think about it: the well-digging process, as it is being modeled by our equation, does not restrict the number of hours to be only whole numbers. Time can indeed be expressed in fractions or decimals. The solution is the number of hours, not the depth, which is why this statement is wrong. It focuses on the well's depth, not the result of the variable h in the equation. In our case, the answer is a whole number, which means that the time can be a decimal, a fraction or an integer. The fact that h comes out to a whole number does not mean that the process could not have included fractional parts of an hour.

Another possible statement might tell us that the solution must be a positive number. This is absolutely true! This is because the context of the problem defines the variable h as time. Time always moves forward; it does not go backward. You cannot dig a well for negative hours. In the equation, a negative sign would imply a reversal in the digging progress, which does not make sense in a well-digging scenario. So, a positive solution is expected and necessary.

Lastly, the options may state something about the solution being even or odd. In this case, since we found that h = 14, an even number, we can conclude that the solution is even. This type of analysis requires us to recognize patterns in numbers, such as even or odd, which might come in handy for other problems. The most critical part of this stage is to understand that the solution is not just an abstract number, but also a meaningful value within the framework of our well-digging scenario. The correct answer would also need to align with our earlier calculations. We need to remember that the equation provides us with a numerical result, and we have to match this to what makes sense in the real world.

Conclusion: The Final Verdict

In conclusion, understanding how to solve the equation is the first step, but it's equally important to interpret the solution in the context of the problem. You need to keep in mind what the variables represent, in this case, the number of hours. If the correct answer says the number of hours must be positive, then you are on the right track. Remember, the solution to the equation is a number, but in this case, it is also a meaningful quantity: the time it takes to dig a well to a certain depth. Therefore, the correct choice should reflect that the number of hours is a positive number and can be an even number. This problem, while seemingly simple, shows us how mathematical equations can be applied to real-world scenarios, and how the interpretation of results is crucial for solving them. Keep practicing, and you'll become a pro at these problems in no time! Keep in mind that math isn't just about finding the answer; it's about making sure that the answer makes sense.