Finding Y When X Is 8 In The Function Y Equals 8 Minus 2x

In the realm of mathematics, functions serve as fundamental tools for modeling relationships between variables. When diving into mathematical functions, understanding how to determine the value of a dependent variable ( extit{y}) for a given independent variable ( extit{x}) is crucial. This article delves into finding the specific value of extit{y} when extit{x} equals 8 in the function y = 8 - 2x. We will explore the process step by step, emphasizing the importance of functions in mathematical problem-solving and their practical applications. By the end of this discussion, you will have a solid grasp of evaluating functions and applying this knowledge to similar scenarios.

The function y = 8 - 2x represents a linear equation, a cornerstone of algebra. Linear functions like this one plot as a straight line on a graph, making them both predictable and easy to work with. In this equation, 8 is the y-intercept (the point where the line crosses the y-axis), and -2 is the slope, indicating that for every increase of 1 in x, y decreases by 2. Understanding this relationship is key to predicting the value of y for any given x. The function notation is a way to express the relationship between an input (x) and an output (y). In this case, the function takes an input x, multiplies it by -2, and then adds 8 to get the output y. This process is fundamental to understanding how functions work and how they can be used to model real-world phenomena. For instance, this function could represent a scenario where you start with 8 units of something and lose 2 units for every unit of time that passes. By understanding the function, we can determine the amount remaining at any given time.

Before we calculate the value of y when x = 8, let's examine the provided table of ordered pairs. Each pair (x, y) represents a point on the line defined by the function. Observing these pairs helps us understand the pattern and behavior of the function. For instance, when x is -3, y is 14, and when x is -1, y is 10. Notice how as x increases, y decreases, which is consistent with the negative slope of the function. This pattern provides a visual confirmation of the function's behavior and helps us anticipate the value of y when x = 8. Analyzing the ordered pairs is a crucial step in understanding the function because it allows us to see the relationship between x and y in a concrete way. Each pair gives us a data point that we can use to verify our calculations and predictions. By carefully examining these pairs, we can gain a deeper understanding of the function's properties and how it behaves over different values of x.

To find the value of y when x = 8, we substitute 8 for x in the equation y = 8 - 2x. This substitution is a fundamental operation in algebra, allowing us to evaluate functions for specific inputs. Once we've substituted, we follow the order of operations (PEMDAS/BODMAS) to simplify the expression. First, we perform the multiplication: 2 * 8 = 16. Then, we subtract this result from 8: 8 - 16 = -8. Therefore, when x = 8, y = -8. This calculation demonstrates how to apply the function rule to determine the output for a given input. The step-by-step approach ensures accuracy and clarity in the solution. By breaking down the calculation into smaller steps, we minimize the risk of errors and make the process easier to understand. This method is applicable to a wide range of functions and is a valuable skill in mathematical problem-solving.

After calculating y = -8 when x = 8, it's essential to verify this result to ensure accuracy. We can do this by comparing our calculated value with the pattern observed in the table of ordered pairs. The table shows a consistent decrease in y as x increases. Our calculated value fits this pattern, providing confidence in our solution. Additionally, we can substitute x = 8 and y = -8 back into the original equation to confirm that the equation holds true. This verification step is crucial in mathematical problem-solving as it helps to identify any potential errors in the calculation process. By taking the time to verify our results, we can ensure that our solutions are accurate and reliable. This practice reinforces our understanding of the function and its behavior, further solidifying our mathematical skills.

In summary, by substituting x = 8 into the function y = 8 - 2x, we determined that y = -8. This exercise demonstrates the practical application of functions in mathematics, highlighting how they allow us to predict the value of a dependent variable for a given independent variable. Understanding and evaluating functions is a crucial skill in mathematics, with applications extending to various fields, including science, engineering, and economics. By mastering these fundamental concepts, we can tackle more complex mathematical problems and gain a deeper appreciation for the power of mathematical modeling. The ability to evaluate functions is not just a theoretical skill; it's a practical tool that enables us to make predictions, analyze data, and solve real-world problems. This article has provided a clear and concise explanation of how to evaluate a linear function, and the principles discussed can be applied to a wide range of mathematical functions.