Solving Inverse Variation: Find V When P = 1/2

by Alex Johnson 47 views

Have you ever wondered how quantities relate when one increases as the other decreases? That's the fascinating world of inverse variation! In this article, we'll explore an inverse variation equation and solve a specific problem. We will break down the concept of inverse variation, walk through the steps to solve the equation, and provide a clear explanation to help you understand the solution. By the end of this article, you'll be equipped to tackle similar problems with confidence. Let's dive in and unravel the mystery of inverse variations together!

Understanding Inverse Variation

Before we jump into solving the problem, let's first understand what inverse variation means. In simple terms, two variables are said to vary inversely if one variable increases as the other decreases, and vice versa. This relationship can be represented mathematically by the equation p = k/V, where p and V are the variables, and k is the constant of variation. The constant of variation is a crucial element in understanding inverse relationships. It represents the fixed value that connects the two inversely related variables. In other words, as one variable changes, the other changes in such a way that their product always equals this constant. Recognizing the constant of variation helps in making predictions and solving problems involving inverse relationships.

In our specific problem, we are given the equation p = 8/V. Here, p and V are the variables, and 8 is the constant of variation. This means that the product of p and V will always be 8. This foundational understanding of inverse variation is essential. It helps us grasp how changes in one variable affect the other. The equation p = k/V is the key to solving a wide array of problems. Whether you're dealing with pressure and volume, speed and time, or any other inversely related quantities, understanding this concept is the first step towards finding solutions. With a firm grasp of inverse variation, we're now ready to tackle the problem at hand: finding the value of V when p is 1/2. Let’s move on to the step-by-step solution to see how this understanding translates into practical problem-solving.

Setting Up the Equation

The problem states that we need to find the value of V when p = 1/2. We are given the inverse variation equation p = 8/V. The first step in solving any mathematical problem is to correctly set up the equation. This involves identifying the given values and substituting them into the appropriate formula. In our case, we know the value of p and we have the equation relating p and V. The next step is to substitute the given value of p into the equation. This will allow us to isolate V and solve for its value.

So, we substitute p = 1/2 into the equation p = 8/V. This gives us 1/2 = 8/V. This substitution is a crucial step because it transforms the general equation into a specific equation with only one unknown variable, which is V. Now that we have the equation 1/2 = 8/V, we can proceed to solve for V. Setting up the equation correctly is more than just plugging in numbers; it's about understanding the relationship between the variables and how they interact within the given context. A clear understanding of this initial setup can make the subsequent steps of solving the problem much smoother and more intuitive. With the equation now set up, we are well-positioned to move forward and find the value of V that satisfies the given conditions. Let’s proceed to the next step where we will solve for V using algebraic manipulation.

Solving for V

Now that we have the equation 1/2 = 8/V, our goal is to isolate V and find its value. To do this, we can use algebraic manipulation. The most straightforward way to solve this equation is to cross-multiply. Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the two products equal to each other. This method is particularly useful for solving equations involving fractions because it eliminates the fractions and simplifies the equation.

So, cross-multiplying in the equation 1/2 = 8/V gives us 1 * V = 8 * 2, which simplifies to V = 16. This means that when p = 1/2, the value of V is 16. Solving for V involves more than just applying a mathematical technique; it requires understanding the properties of equations and how to manipulate them to isolate the unknown variable. Each step in the process is a logical progression, building upon the previous step to ultimately arrive at the solution. With V now isolated, we have successfully found its value. But our work isn't quite done yet. It's important to verify our solution to ensure it makes sense within the context of the original problem. In the next section, we'll discuss how to verify the solution and confirm that our answer is correct. Let’s move on to the verification step to ensure the accuracy of our solution.

Verifying the Solution

After solving for a variable, it's always a good practice to verify the solution. This ensures that the calculated value is correct and satisfies the original equation. To verify our solution, we substitute the value we found for V back into the original equation and check if the equation holds true. In our case, we found that V = 16 when p = 1/2. The original equation is p = 8/V. So, we substitute V = 16 into the equation to see if it holds true for p = 1/2.

Substituting V = 16 into p = 8/V gives us p = 8/16, which simplifies to p = 1/2. This matches the given value of p, so our solution is verified. Verification is a critical step in problem-solving because it helps catch any errors that might have occurred during the calculation process. By substituting the solution back into the original equation, we can confirm that our answer is consistent with the given information and the relationships between the variables. This step adds an extra layer of confidence in our solution. Knowing how to verify a solution is a valuable skill. It’s not just about finding an answer; it’s about ensuring that the answer is correct and makes sense in the context of the problem. Now that we have verified our solution, we can confidently say that when p = 1/2, the value of V is indeed 16. With the solution verified, let's move on to a summary of the key steps we took to solve this problem, reinforcing the concepts we've covered.

Conclusion

In this article, we successfully solved an inverse variation problem. We were given the equation p = 8/V and asked to find the value of V when p = 1/2. We started by understanding the concept of inverse variation, which states that as one variable increases, the other decreases proportionally. We then set up the equation by substituting the given value of p into the equation. Next, we solved for V using algebraic manipulation, specifically cross-multiplication. We found that V = 16. Finally, we verified our solution by substituting V = 16 back into the original equation and confirming that it holds true for p = 1/2.

Understanding inverse variation is crucial in various real-world applications, from physics to economics. The ability to set up and solve these types of equations is a valuable skill. Whether you're a student learning algebra or someone looking to refresh their math skills, mastering inverse variation can open doors to solving a wide range of problems. Remember, the key to solving inverse variation problems is to understand the relationship between the variables, set up the equation correctly, and use algebraic techniques to isolate the unknown variable. And always remember to verify your solution! By following these steps, you'll be well-equipped to tackle any inverse variation problem that comes your way. For further learning on mathematical concepts and problem-solving, you can visit resources like Khan Academy.