Solving Direct Variation Problems Finding Y When X Is -6
In the realm of mathematics, understanding direct variation is crucial for solving a wide range of problems. Direct variation, at its core, describes a relationship between two variables where one variable is a constant multiple of the other. This means as one variable increases, the other increases proportionally, and as one decreases, the other decreases proportionally. This fundamental concept is applied in various fields, from physics and engineering to economics and everyday life. When dealing with direct variation problems, it is essential to have a systematic approach to identify the relationship between the variables, determine the constant of variation, and use this information to solve for unknown values. This article aims to provide a comprehensive guide on how to solve direct variation problems, using a specific example to illustrate the process. We will break down the problem step-by-step, explaining the underlying principles and techniques involved. By the end of this article, you will have a solid understanding of direct variation and be well-equipped to tackle similar problems with confidence. Direct variation is not merely an abstract mathematical concept; it has practical applications in various real-world scenarios. For example, the relationship between the distance traveled and time taken at a constant speed is a direct variation. Similarly, the relationship between the number of items purchased and the total cost (assuming a fixed price per item) also demonstrates direct variation. Understanding this relationship allows us to make predictions and solve problems in everyday situations. Therefore, mastering direct variation is not only beneficial for academic purposes but also for developing practical problem-solving skills. This article will provide you with a clear and concise explanation of direct variation, enabling you to apply this concept effectively in different contexts. We will explore the key characteristics of direct variation, the methods for identifying and representing it mathematically, and the steps involved in solving problems related to direct variation. By focusing on a specific example, we will demonstrate how to approach these problems systematically and arrive at the correct solution. Let's delve into the world of direct variation and unlock its potential for problem-solving.
Understanding Direct Variation
Direct variation, a fundamental concept in algebra, describes a relationship between two variables where one variable is a constant multiple of the other. This relationship can be expressed mathematically as y = kx, where y and x are the variables, and k is the constant of variation. The constant of variation, k, represents the factor by which x must be multiplied to obtain y. In simpler terms, if y varies directly with x, it means that as x changes, y changes proportionally. For instance, if x doubles, y also doubles, and if x halves, y also halves. This proportional relationship is the defining characteristic of direct variation. Understanding the concept of direct variation is crucial for solving problems in various fields, including physics, engineering, and economics. Many real-world phenomena exhibit direct variation, such as the relationship between distance and time at a constant speed, or the relationship between the number of workers and the amount of work completed in a fixed time. Identifying direct variation in a given situation allows us to establish a mathematical model that accurately represents the relationship between the variables. This model can then be used to make predictions and solve for unknown values. One of the key aspects of direct variation is the constant of variation, k. This constant represents the slope of the line when the direct variation relationship is graphed on a coordinate plane. The value of k determines the steepness of the line, and its sign indicates whether the relationship is increasing (positive k) or decreasing (negative k). Determining the constant of variation is often the first step in solving direct variation problems. This can be done by using a given set of values for x and y and substituting them into the equation y = kx. Once k is known, the equation can be used to find the value of y for any given value of x, or vice versa. In this article, we will delve deeper into the process of solving direct variation problems, using a specific example to illustrate the steps involved. By understanding the underlying principles and applying a systematic approach, you can master direct variation and confidently tackle a wide range of problems.
Problem Statement: If there is direct variation and y=-5 when x=2, find y when x=-6
The problem at hand is a classic example of a direct variation problem. The statement explicitly tells us that there is direct variation between two variables, y and x. This immediately informs us that the relationship between y and x can be expressed in the form y = kx, where k is the constant of variation. The problem provides us with an initial set of values: y = -5 when x = 2. This information is crucial because it allows us to determine the value of the constant of variation, k. By substituting these values into the equation y = kx, we can solve for k and establish the specific relationship between y and x for this particular problem. Once we have determined the value of k, we can then use the direct variation equation to find the value of y for any given value of x. The problem asks us to find the value of y when x = -6. This is the second part of the problem, where we apply the direct variation equation with the known value of k and the given value of x to solve for y. Understanding the problem statement is the first and most important step in solving any mathematical problem. In this case, the statement clearly indicates the presence of direct variation and provides the necessary information to determine the constant of variation. By carefully analyzing the problem statement, we can break it down into smaller, more manageable steps. The first step is to use the given values of y and x to find k. The second step is to use the calculated value of k and the new value of x to find the corresponding value of y. This systematic approach ensures that we address all aspects of the problem and arrive at the correct solution. In the following sections, we will delve into the detailed steps of solving this problem, providing a clear and concise explanation of each step. By understanding the process involved in solving this problem, you will be well-equipped to tackle similar direct variation problems with confidence.
Step-by-Step Solution
To solve the direct variation problem, we need to follow a step-by-step approach. As highlighted earlier, the key to unlocking direct variation problems lies in the formula y = kx, where k is the constant of variation.
Step 1: Find the Constant of Variation (k)
We are given that y = -5 when x = 2. We can substitute these values into the direct variation equation to solve for k:
-5 = k(2)
To isolate k, we divide both sides of the equation by 2:
k = -5 / 2
k = -2.5
Therefore, the constant of variation, k, is -2.5. This means that y is -2.5 times x. Understanding the constant of variation is crucial as it defines the relationship between the two variables. In this case, a negative constant indicates that as x increases, y decreases, and vice versa. This inverse relationship is an important characteristic to consider when analyzing the problem and interpreting the results. The constant of variation acts as a bridge between x and y, allowing us to calculate one variable if we know the other. It is a fundamental concept in direct variation and must be determined accurately to solve the problem correctly. Now that we have found the value of k, we can move on to the next step, which is to use this value to find y when x = -6.
Step 2: Find y when x = -6
Now that we have the constant of variation, k = -2.5, we can use the direct variation equation to find the value of y when x = -6. We substitute the values of k and x into the equation:
y = -2.5 * (-6)
Multiplying -2.5 by -6, we get:
y = 15
Therefore, when x = -6, y = 15. This means that based on the direct variation relationship established by the given information, the corresponding value of y is 15. This result is consistent with the negative constant of variation, which indicates an inverse relationship between x and y. As x changes from a positive value (2) to a negative value (-6), y changes from a negative value (-5) to a positive value (15). This understanding of the relationship between the variables is crucial for verifying the correctness of the solution. In this step, we have successfully applied the concept of direct variation to find the unknown value of y. By using the constant of variation and the given value of x, we have solved the problem and arrived at the answer. This demonstrates the power of direct variation in solving problems involving proportional relationships between variables. With this solution, we have completed the step-by-step process of solving the given problem. In the next section, we will summarize the solution and discuss the implications of the result.
Solution and Answer
Having meticulously followed the steps outlined above, we arrive at the solution to the direct variation problem. In the first step, we determined the constant of variation, k, to be -2.5 by substituting the given values of y = -5 and x = 2 into the direct variation equation y = kx. This constant, -2.5, signifies the factor by which x is multiplied to obtain y, and it plays a pivotal role in defining the relationship between the two variables. Moving on to the second step, we utilized the calculated value of k and the new value of x = -6 to find the corresponding value of y. By substituting these values into the equation y = kx, we obtained y = -2.5 * (-6), which simplifies to y = 15. Therefore, when x = -6, y = 15. This is the solution to the problem, answering the question of what the value of y is when x equals -6. The solution, y = 15, is one of the options provided in the problem statement. Specifically, it corresponds to option C. This confirms that our step-by-step approach has led us to the correct answer. It is important to note that the negative constant of variation indicates an inverse relationship between x and y. As x changes from a positive value to a negative value, y changes from a negative value to a positive value. This understanding of the relationship between the variables helps to verify the correctness of the solution and provides a deeper insight into the problem. In summary, the solution to the direct variation problem is y = 15 when x = -6, which corresponds to option C. This solution has been obtained through a systematic approach, involving the determination of the constant of variation and the application of the direct variation equation. By understanding the principles of direct variation and following a step-by-step method, we can confidently solve similar problems and apply this knowledge to various real-world scenarios.
The final answer is C. 15.
Conclusion
In conclusion, this article has provided a comprehensive guide on solving direct variation problems, using a specific example to illustrate the process. We began by defining direct variation and explaining its mathematical representation as y = kx, where k is the constant of variation. Understanding the concept of direct variation is crucial for solving problems in various fields, as it describes a proportional relationship between two variables. We then presented the problem statement, which involved finding the value of y when x = -6, given that y = -5 when x = 2 and that there is direct variation between y and x. By carefully analyzing the problem statement, we identified the key information and outlined a step-by-step approach to solve the problem. The first step in our solution was to find the constant of variation, k. We achieved this by substituting the given values of y and x into the direct variation equation and solving for k. This resulted in k = -2.5, which signifies the factor by which x is multiplied to obtain y. The second step was to use the calculated value of k and the new value of x = -6 to find the corresponding value of y. By substituting these values into the equation y = kx, we obtained y = 15. Therefore, when x = -6, y = 15. This is the solution to the problem, and it corresponds to option C in the given choices. Throughout the solution process, we emphasized the importance of understanding the underlying principles of direct variation and applying a systematic approach. By breaking down the problem into smaller, more manageable steps, we were able to solve it effectively and accurately. Furthermore, we highlighted the significance of the constant of variation and its role in defining the relationship between the variables. The negative constant in this example indicates an inverse relationship, which is an important characteristic to consider when interpreting the results. In summary, this article has provided a clear and concise explanation of how to solve direct variation problems. By understanding the concept, applying a step-by-step approach, and paying attention to the details, you can confidently tackle similar problems and apply this knowledge to various real-world scenarios. The key takeaways from this article are the understanding of direct variation, the determination of the constant of variation, and the application of the direct variation equation to solve for unknown values. With these skills, you are well-equipped to excel in solving direct variation problems.
