Direct Variation Problem Solving Y When X Is 30

Direct variation is a fundamental concept in mathematics that describes a relationship between two variables where one variable is a constant multiple of the other. In simpler terms, as one variable increases, the other variable increases proportionally, and vice versa. This relationship can be represented mathematically by the equation y = kx, where y and x are the variables, and k is the constant of variation. The constant of variation, denoted by k, represents the factor by which x must be multiplied to obtain y. It essentially determines the steepness of the relationship between the two variables. Understanding direct variation is crucial in various fields, including physics, engineering, and economics, as it allows us to model and predict how quantities change in relation to each other. For example, the distance traveled by a car at a constant speed varies directly with the time traveled, and the amount of money earned varies directly with the number of hours worked at a fixed hourly rate.

To solidify your understanding of direct variation, consider the following examples: If you are paid an hourly wage, the amount you earn varies directly with the number of hours you work. The more hours you work, the more you earn, assuming your hourly rate remains constant. The distance a car travels at a constant speed varies directly with the time spent driving. If you double the driving time, you double the distance covered, provided the speed remains the same. The circumference of a circle varies directly with its diameter. The circumference is always π (pi) times the diameter, which means the constant of variation is π. These examples illustrate the practical applications of direct variation in everyday scenarios. Recognizing and understanding these relationships can help you solve problems and make informed decisions in various contexts. Understanding direct variation is not just about memorizing the formula y = kx; it's about grasping the concept of proportionality and how it manifests in real-world situations. Once you understand the underlying principle, you can apply it to a wide range of problems and gain a deeper appreciation for the interconnectedness of mathematical concepts.

In this specific problem, we are presented with a direct variation scenario where the variable y varies directly with the variable x. This means that there is a constant k such that y = kx. We are given an initial condition: when x = 20, y = 80. This information is crucial because it allows us to determine the constant of variation, k. Once we find k, we can use the direct variation equation to find the value of y for any given value of x. The problem then asks us to find the value of y when x = 30. This is a typical direct variation problem that requires us to first find the constant of variation and then use it to calculate the value of the dependent variable (y) for a new value of the independent variable (x). This type of problem is a common application of direct variation and highlights the importance of understanding how to manipulate the equation y = kx to solve for different variables.

The problem statement is concise and clearly outlines the relationship between x and y. It provides the necessary information to solve the problem, including the initial condition (y = 80 when x = 20) and the target value of x (x = 30). By understanding the direct variation concept and applying the given information, we can systematically solve for the unknown value of y. The problem's structure is straightforward, making it an excellent example for illustrating the steps involved in solving direct variation problems. It emphasizes the importance of identifying the constant of variation as the key to unlocking the relationship between the variables. The problem also implicitly reinforces the idea that direct variation represents a linear relationship, where the graph of y versus x would be a straight line passing through the origin. This visual representation can further enhance understanding and provide an alternative approach to solving the problem. Overall, the problem statement is well-defined and serves as a solid foundation for applying the principles of direct variation.

To begin solving this direct variation problem, our first task is to determine the constant of variation, k. Recall that the direct variation equation is y = kx. We are given that y = 80 when x = 20. We can substitute these values into the equation to solve for k. This substitution gives us the equation 80 = k * 20. To isolate k, we need to divide both sides of the equation by 20. This gives us k = 80 / 20, which simplifies to k = 4. Therefore, the constant of variation in this case is 4. This means that y is always 4 times x. The constant of variation is a crucial element in direct variation problems because it defines the specific relationship between the two variables. Once we know k, we can accurately predict the value of y for any given value of x, and vice versa. In essence, k acts as a scaling factor that determines how much y changes for each unit change in x.

Finding the constant of variation is often the first and most important step in solving direct variation problems. It allows us to establish a concrete relationship between the variables, which is essential for making accurate predictions and solving for unknowns. The process of finding k involves substituting the given values of x and y into the direct variation equation and then using algebraic manipulation to isolate k. This step reinforces the importance of understanding and applying algebraic principles in problem-solving. The value of k also provides insight into the nature of the relationship between x and y. A larger value of k indicates a steeper relationship, meaning that y changes more rapidly with respect to x. Conversely, a smaller value of k indicates a less steep relationship. In this specific problem, k = 4 tells us that for every unit increase in x, y increases by 4 units. This understanding of the constant of variation helps us to interpret the direct variation relationship in a meaningful way and to apply it to various contexts.

Now that we have determined the constant of variation, k = 4, we can proceed to the next step: calculating the value of y when x = 30. We will use the direct variation equation, y = kx, and substitute the values of k and x that we know. Substituting k = 4 and x = 30 into the equation, we get y = 4 * 30. Performing the multiplication, we find that y = 120. Therefore, when x = 30, the value of y is 120. This calculation demonstrates how the constant of variation allows us to easily find the value of one variable when we know the value of the other variable in a direct variation relationship. Once we have established the relationship between x and y through the constant of variation, we can use the equation y = kx to solve for any unknown value.

This step highlights the practical application of direct variation. By finding the constant of variation, we have essentially created a rule that governs the relationship between x and y. This rule allows us to make predictions and solve problems involving these variables. The calculation itself is straightforward, but it is important to understand the underlying concept of direct variation and how the constant of variation connects the two variables. The result, y = 120, provides a specific answer to the problem's question. It demonstrates that as x increases from 20 to 30, y also increases proportionally, as dictated by the constant of variation. This proportional increase is a key characteristic of direct variation and is what makes it a useful tool for modeling and understanding real-world relationships. The ability to calculate y for a given x using the direct variation equation is a fundamental skill in mathematics and has applications in various fields, including science, engineering, and economics.

In conclusion, by following the steps of identifying the direct variation relationship, finding the constant of variation, and substituting the given value of x, we have successfully determined the value of y when x = 30. The final answer is y = 120. This result demonstrates the power of understanding and applying the concept of direct variation to solve mathematical problems. The process of solving this problem involved a combination of algebraic manipulation and conceptual understanding, highlighting the importance of both in mathematics. The answer, y = 120, is a specific solution to the given problem, but the underlying principles and techniques used to arrive at this answer are applicable to a wide range of direct variation problems. This type of problem serves as a fundamental building block for more advanced mathematical concepts and applications.

The problem-solving approach used in this example can be generalized to any direct variation problem. The key is to first recognize the direct variation relationship, then use the given information to find the constant of variation, and finally use the equation y = kx to solve for the unknown variable. This systematic approach ensures accuracy and efficiency in problem-solving. The final answer, y = 120, not only provides a numerical solution but also reinforces the understanding of the proportional relationship between x and y. As x increases, y increases proportionally, and this relationship is governed by the constant of variation. The successful solution of this problem demonstrates the importance of mathematical reasoning and the ability to apply mathematical concepts to real-world scenarios. Direct variation is a fundamental concept in mathematics, and mastering it is essential for success in more advanced topics.

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