Solving Inverse Variation Problems Finding Y When X Is -3

Inverse variation, a fundamental concept in mathematics, describes a relationship between two variables where an increase in one variable results in a proportional decrease in the other. This relationship is mathematically expressed as $y = k/x$, where 'y' and 'x' are the variables, and 'k' is the constant of variation. Understanding inverse variation is crucial in various fields, including physics, engineering, and economics, as it helps model real-world scenarios where quantities are inversely related. For example, the time it takes to travel a certain distance is inversely proportional to the speed of travel; as speed increases, travel time decreases, assuming the distance remains constant. Similarly, the pressure of a gas in a closed container is inversely proportional to its volume, as described by Boyle's Law. In graphical terms, inverse variation is represented by a hyperbola, a curve with two branches that approach but never touch the axes. This unique shape visually demonstrates the inverse relationship between the variables. Recognizing and applying the principles of inverse variation allows for solving a wide range of problems, from determining the appropriate gear ratios in machines to predicting the behavior of economic markets. The constant of variation, 'k', plays a pivotal role in defining the specific relationship between the variables. It essentially scales the inverse relationship, dictating how much 'y' changes for a given change in 'x'. To find 'k', you typically need a known pair of 'x' and 'y' values, which can then be substituted into the equation $y = k/x$. Once 'k' is determined, the equation can be used to predict 'y' for any given 'x', and vice versa. This predictive power is what makes inverse variation such a valuable tool in mathematical modeling and problem-solving. The concept of inverse variation is not limited to simple scenarios with two variables. It can be extended to more complex situations involving multiple variables and more intricate relationships. For instance, the gravitational force between two objects is inversely proportional to the square of the distance between them, illustrating an inverse square relationship. These more advanced applications demonstrate the versatility and widespread applicability of inverse variation in understanding and modeling the world around us.

Problem Statement: Unveiling the Solution

The problem at hand presents a classic scenario involving inverse variation. We are given that $y$ varies inversely with $x$, and we have an initial condition: when $x = -2$, $y = 6$. Our objective is to find the value of $y$ when $x = -3$. This problem exemplifies how inverse variation principles can be applied to solve practical problems. To begin, we recognize the fundamental equation that governs inverse variation: $y = k/x$, where $k$ is the constant of variation. Our first task is to determine the value of $k$ using the given initial condition. Substituting $x = -2$ and $y = 6$ into the equation, we get $6 = k/(-2)$. Solving for $k$, we multiply both sides of the equation by -2, which gives us $k = -12$. Now that we have found the constant of variation, we can write the specific equation for this particular inverse relationship as $y = -12/x$. This equation allows us to calculate the value of $y$ for any given value of $x$. The next step is to find $y$ when $x = -3$. We substitute $x = -3$ into our equation, yielding $y = -12/(-3)$. Performing the division, we find that $y = 4$. Therefore, when $x = -3$, the value of $y$ is 4. This result demonstrates the inverse nature of the relationship; as $x$ changes from -2 to -3, the value of $y$ changes from 6 to 4, reflecting the inverse proportionality. The ability to solve such problems is crucial in understanding and applying the principles of inverse variation. These types of problems often appear in various contexts, from physics to economics, where understanding the relationships between variables is essential. The methodical approach of first finding the constant of variation and then using it to solve for the unknown variable is a key strategy in tackling inverse variation problems. By mastering this approach, one can confidently address a wide range of scenarios involving inversely proportional quantities.

Step-by-Step Solution: Cracking the Code

To solve this inverse variation problem, we follow a structured, step-by-step approach. First, we acknowledge the fundamental equation that governs inverse variation: $y = k/x$, where $y$ and $x$ are the variables, and $k$ is the constant of variation. The first crucial step is to determine the value of this constant, $k$. We are given an initial condition: when $x = -2$, $y = 6$. We substitute these values into the inverse variation equation: $6 = k/(-2)$. To isolate $k$, we multiply both sides of the equation by -2: $6 * (-2) = k$. This simplifies to $k = -12$. Now that we have found the constant of variation, $k = -12$, we can write the specific equation representing this inverse relationship: $y = -12/x$. This equation is the key to solving the problem, as it relates $y$ and $x$ directly for this particular scenario. The next step is to use this equation to find the value of $y$ when $x = -3$. We substitute $x = -3$ into the equation: $y = -12/(-3)$. Performing the division, we get $y = 4$. Thus, when $x = -3$, the value of $y$ is 4. This result confirms the inverse relationship; as $x$ changes from -2 to -3, $y$ changes from 6 to 4. The negative signs also play a crucial role in the calculation. Dividing a negative number by a negative number results in a positive number, which is why $y$ is positive when $x$ is negative. This step-by-step approach highlights the importance of first finding the constant of variation, as it allows us to establish the specific relationship between the variables. Once the constant is known, we can easily solve for any unknown variable given the value of the other. This methodical approach is applicable to a wide range of inverse variation problems and provides a clear and concise way to arrive at the solution. Mastering this technique is essential for anyone studying inverse variation and its applications.

Identifying the Correct Answer: The Final Verdict

After performing the calculations, we have determined that when $x = -3$, the value of $y$ is 4. Now, we need to identify which of the provided options matches our solution. The options are:

A. 1 B. -4 C. 4 D. -1

Comparing our calculated value of $y = 4$ with the given options, we can clearly see that option C, 4, is the correct answer. Options A, B, and D are incorrect as they do not match our calculated value. This step of verifying the solution against the provided options is crucial in problem-solving. It ensures that we not only perform the calculations correctly but also accurately identify the corresponding answer. Sometimes, even if the calculations are correct, a mistake in selecting the answer choice can lead to an incorrect result. Therefore, taking the time to carefully compare the calculated value with the options is a vital step in the problem-solving process. In this case, the correct answer, option C, aligns perfectly with our calculated value of $y = 4$. This confirms that we have correctly applied the principles of inverse variation and solved the problem accurately. The process of elimination can also be helpful in identifying the correct answer. If we had initially obtained a different value for $y$, we could have eliminated the options that did not match our calculated value, narrowing down the possibilities. This strategy is particularly useful when dealing with multiple-choice questions, as it can help increase the chances of selecting the correct answer, even if there is some uncertainty in the calculations. The final step of identifying the correct answer is just as important as the calculations themselves. It represents the culmination of the problem-solving process and ensures that we arrive at the correct solution. By carefully verifying our answer against the provided options, we can confidently conclude that we have successfully solved the problem.

Conclusion: Mastering Inverse Variation

In conclusion, we have successfully solved the problem involving inverse variation. We were given that $y = 6$ when $x = -2$, and we were tasked with finding $y$ when $x = -3$. By applying the principles of inverse variation, we determined that the correct answer is $y = 4$. This problem highlights the importance of understanding the fundamental equation of inverse variation, $y = k/x$, and the significance of the constant of variation, $k$. To solve the problem, we first calculated the constant of variation using the given initial condition. Substituting $x = -2$ and $y = 6$ into the equation, we found that $k = -12$. This allowed us to write the specific equation for this inverse relationship as $y = -12/x$. Next, we substituted $x = -3$ into this equation to find the corresponding value of $y$. This gave us $y = -12/(-3)$, which simplifies to $y = 4$. Therefore, when $x = -3$, $y = 4$. This result demonstrates the inverse relationship between $x$ and $y$; as $x$ decreases from -2 to -3, $y$ increases from 6 to 4. The negative signs also play a crucial role, as dividing a negative number by a negative number results in a positive number. The step-by-step approach we followed in solving this problem is applicable to a wide range of inverse variation problems. This approach involves first finding the constant of variation, then using it to establish the specific equation, and finally substituting the given value to solve for the unknown variable. Mastering this technique is essential for anyone studying inverse variation and its applications in various fields, including physics, engineering, and economics. Inverse variation is a fundamental concept in mathematics, and understanding its principles is crucial for solving real-world problems involving inversely proportional quantities. By practicing and applying these principles, one can confidently tackle various scenarios involving inverse variation.