Solving Direct Proportionality Problems Finding Y When X Is -2

Direct proportionality is a fundamental concept in mathematics that describes a relationship between two variables where one variable changes in direct proportion to another. In simpler terms, if one variable increases, the other variable increases proportionally, and if one variable decreases, the other variable decreases proportionally. This relationship can be expressed mathematically as y = kx, where y and x are the variables, and k is a constant of proportionality. This constant represents the factor by which y changes for every unit change in x. Understanding direct proportionality is crucial for solving various mathematical problems and for understanding real-world phenomena where quantities vary in relation to each other. For instance, the distance traveled by a car at a constant speed is directly proportional to the time traveled, and the cost of buying a certain number of items at a fixed price per item is directly proportional to the number of items purchased. To effectively work with direct proportionality, it's essential to identify the variables involved, determine the constant of proportionality, and then use this constant to find the value of one variable given the value of the other. This involves setting up equations, substituting known values, and solving for the unknown variable. The principles of direct proportionality extend to scenarios where the relationship involves powers of variables, such as y being directly proportional to x² or x³, requiring a slight modification in the mathematical representation but maintaining the core concept of proportional change.

In this particular problem, y is directly proportional to x², which means that the value of y changes proportionally to the square of x. This can be expressed as y = kx², where k is the constant of proportionality. This relationship implies that as x increases, y increases at a faster rate, specifically at a rate proportional to the square of x. Understanding this relationship is crucial for solving the problem, as we need to determine the value of k to find the specific equation that relates y and x. To find the value of k, we are given additional information: the difference in the values of y when x = 1 and x = 3 is 32. This information provides us with a specific scenario that we can use to create equations and solve for k. By setting up equations based on the given information, we can substitute the values of x and the difference in y to isolate k and find its value. Once we have the value of k, we can use the equation y = kx² to find the value of y for any given value of x, including the target value of x = -2. Therefore, the key to solving this problem lies in understanding the direct proportionality relationship, using the given information to determine the constant of proportionality, and then applying the derived equation to find the desired value of y. The problem demonstrates a common application of direct proportionality in mathematical problem-solving, where understanding the relationship between variables and using given information strategically are essential for arriving at the correct solution.

To determine the constant of proportionality, k, we need to use the information provided about the difference in the values of y when x = 1 and x = 3. Let's denote the value of y when x = 1 as y₁ and the value of y when x = 3 as y₂. According to the direct proportionality relationship y = kx², we can write two equations:

• y₁ = k(1)² = k
• y₂ = k(3)² = 9k

We are given that the difference in the values of y is 32, which means y₂ - y₁ = 32. Substituting the expressions for y₁ and y₂ from the equations above, we get:

• 9k - k = 32

Simplifying the equation, we have:

• 8k = 32

Dividing both sides by 8, we find the value of k:

• k = 4

Therefore, the constant of proportionality, k, is 4. This means that the specific equation relating y and x in this problem is y = 4x². This constant is crucial because it establishes the exact relationship between x and y, allowing us to calculate the value of y for any given value of x. With the value of k determined, we can now proceed to the final step of the problem, which is to find the value of y when x = -2. Understanding how to calculate the constant of proportionality is a key skill in solving problems involving direct proportionality, as it forms the foundation for determining the specific relationship between the variables.

Now that we have determined the constant of proportionality, k = 4, we can use the equation y = 4x² to find the value of y when x = -2. To do this, we simply substitute x = -2 into the equation:

• y = 4(-2)²

First, we calculate the square of -2:

• (-2)² = 4

Then, we multiply the result by 4:

• y = 4 * 4 = 16

Therefore, the value of y when x = -2 is 16. This result demonstrates how the direct proportionality relationship, once established with the constant of proportionality, can be used to easily find the value of one variable given the value of the other. The process of substituting the value of x into the equation and performing the calculations is straightforward, but it relies on the initial determination of k. The value of y being 16 when x = -2 fits the direct proportionality relationship y = 4x², as the square of -2 is 4, and multiplying that by the constant 4 gives us 16. This final step completes the problem and provides the answer to the question posed. The entire solution process highlights the importance of understanding direct proportionality, setting up equations based on given information, solving for unknowns, and applying the derived relationship to find specific values.

In conclusion, by understanding the principles of direct proportionality and utilizing the given information effectively, we have successfully determined that the value of y when x = -2 is 16. The problem presented a scenario where y is directly proportional to x², and we were tasked with finding the value of y for a specific value of x. The solution involved several key steps. First, we established the general equation for the direct proportionality relationship, y = kx², where k is the constant of proportionality. Then, we used the information about the difference in the values of y when x = 1 and x = 3 to set up equations and solve for k. By calculating the difference in y values and equating it to 32, we found that k = 4. This constant established the specific relationship between y and x in this problem as y = 4x². Finally, with the constant of proportionality determined, we substituted x = -2 into the equation to find the corresponding value of y. After performing the calculations, we found that y = 16 when x = -2. This solution demonstrates the power of direct proportionality as a mathematical concept and its applicability in solving problems involving related variables. The ability to identify direct proportionality, set up equations, solve for constants, and apply the resulting relationships is a valuable skill in mathematics and various other fields. The problem also illustrates the importance of breaking down complex problems into smaller, manageable steps, which makes the solution process more organized and easier to follow.

This problem provides several key takeaways for understanding and working with direct proportionality relationships. Firstly, it emphasizes the importance of recognizing and understanding the fundamental concept of direct proportionality. The relationship y = kx² signifies that y changes in proportion to the square of x, and understanding this relationship is crucial for setting up the problem correctly. Secondly, the problem highlights the significance of the constant of proportionality, k. This constant is the key to establishing the specific relationship between the variables, and it must be determined using the given information. In this case, the information about the difference in y values allowed us to solve for k. Thirdly, the problem demonstrates the process of translating word problems into mathematical equations. The ability to convert the given information into equations is a fundamental skill in mathematics, and this problem provides a clear example of how to do so. Fourthly, the solution process showcases the step-by-step approach to problem-solving. By breaking the problem down into smaller steps, such as finding k first and then solving for y, the solution becomes more manageable. Finally, the problem reinforces the importance of accuracy in calculations. A single error in calculation can lead to an incorrect answer, so it's crucial to double-check each step. These takeaways are valuable for anyone studying mathematics or working with problems involving direct proportionality. By mastering these concepts and skills, one can confidently approach and solve a wide range of mathematical problems.

Direct proportionality is not just a theoretical concept; it has numerous real-world applications across various fields. Understanding direct proportionality helps us model and analyze situations where quantities vary in relation to each other. In physics, for example, the distance traveled by an object moving at a constant speed is directly proportional to the time traveled. This relationship is expressed by the equation d = vt, where d is distance, v is speed, and t is time. Similarly, in economics, the cost of purchasing a certain number of items at a fixed price per item is directly proportional to the number of items purchased. If the price per item is p and the number of items is n, the total cost C is given by C = pn. In chemistry, the amount of product formed in a chemical reaction is often directly proportional to the amount of reactants used, assuming sufficient conditions for the reaction to proceed. In engineering, the elongation of a spring is directly proportional to the force applied to it, as described by Hooke's Law. This principle is used in various mechanical systems and structures. Direct proportionality also finds applications in everyday life. For instance, the amount of ingredients needed for a recipe is directly proportional to the number of servings being prepared. If a recipe calls for certain amounts of ingredients for 4 servings, doubling the ingredients will yield 8 servings. These examples illustrate the widespread applicability of direct proportionality in understanding and modeling real-world phenomena. By recognizing and applying the principles of direct proportionality, we can solve a variety of practical problems and make informed decisions in various contexts.

When solving problems involving direct proportionality, it's important to be aware of common mistakes that can lead to incorrect answers. One frequent mistake is not correctly identifying the direct proportionality relationship. For example, in the given problem, y is directly proportional to x², not x. Failing to recognize this can lead to setting up the equation incorrectly. Another common mistake is an incorrect calculation of the constant of proportionality, k. This constant is crucial for establishing the specific relationship between the variables, and any error in its calculation will propagate through the rest of the solution. It's essential to carefully check the equations used to solve for k and ensure that the substitutions and algebraic manipulations are accurate. A third mistake is incorrect substitution of values into the equation. When finding the value of one variable given the value of the other, it's important to substitute the correct values into the correct places in the equation. A misplaced value can result in a wrong answer. Another error can arise from misinterpreting the problem statement. It's crucial to carefully read and understand the problem before attempting to solve it. Misinterpreting the given information or the question being asked can lead to an incorrect approach. Finally, careless arithmetic errors can also lead to mistakes. Even if the problem-solving process is correct, a simple addition or multiplication error can result in an incorrect answer. It's advisable to double-check all calculations to avoid such errors. By being aware of these common mistakes and taking steps to avoid them, one can increase the accuracy and efficiency of solving direct proportionality problems.