Finding Y Value: Function Y=8-2x When X=8

Hey guys! Today, we're diving into the world of functions and ordered pairs. We've got a table showing some ordered pairs for the function y = 8 - 2x, and our mission is to find the value of y when x is 8. Sounds like a fun puzzle, right? Let's get started!

Understanding the Function and Ordered Pairs

Before we jump into solving the problem, let's make sure we're all on the same page about functions and ordered pairs. In simple terms, a function is like a machine that takes an input, does something to it, and spits out an output. In our case, the input is x, the machine is the equation y = 8 - 2x, and the output is y.

Ordered pairs are simply pairs of numbers (x, y) that represent a specific point on a graph. They tell us the x-coordinate and the y-coordinate of that point. The table we have shows us a few of these points for our function. For example, the first row tells us that when x is -3, y is 14. This means the point (-3, 14) lies on the graph of our function.

Think of it like a map. The x-value tells you how far to move horizontally, and the y-value tells you how far to move vertically. Together, they pinpoint a specific location – a point on the graph. Understanding this relationship between x, y, and the function is key to solving our problem.

The beauty of a function like y = 8 - 2x is that it provides a rule or a relationship that connects every x-value to a unique y-value. This rule allows us to predict what y will be for any given x. That's precisely what we're going to do now: find the y that corresponds to x = 8. We will use this function to find the missing y value when x equals 8. This involves substituting 8 for x in the equation and solving for y. This is a straightforward application of the function's rule. By understanding the function and how ordered pairs represent points, we can easily solve for the unknown y-value. Let's move on to the next section where we'll actually do the calculation. Remember, math is all about understanding the relationships and applying the rules. Once you get the hang of it, it becomes a powerful tool for solving all sorts of problems!

Solving for y When x = 8

Alright, let's get down to business! We know our function is y = 8 - 2x, and we want to find y when x is 8. The key here is substitution. We're going to replace the x in our equation with the value 8.

So, here's how it looks:

y = 8 - 2(8)*

Now, we need to follow the order of operations (PEMDAS/BODMAS) to simplify this expression. First up, we do the multiplication:

y = 8 - 16

And finally, we do the subtraction:

y = -8

Boom! We've found our answer. When x is 8, y is -8. That wasn't so bad, was it? This process of substituting and simplifying is a fundamental skill in algebra, and you'll use it all the time.

Think of substitution as plugging in a value. We're taking the information we know (x = 8) and plugging it into the equation to find the information we don't know (y). It's like filling in the blanks in a puzzle. Each step we took, from substituting to multiplying to subtracting, was crucial in arriving at the correct answer. Skipping a step or doing them in the wrong order could lead to mistakes. That's why understanding the order of operations is so important. Moreover, this simple calculation highlights the direct relationship defined by the function. For each input x, the function precisely determines the output y. Let's take a closer look at how this solution fits within the context of the table provided in the problem. This will help us to verify our answer and ensure that it aligns with the pattern established by the other ordered pairs. Next, we'll see how our answer fits into the bigger picture.

Verifying the Solution in the Table

Now that we've calculated y = -8 when x = 8, let's make sure our answer makes sense in the context of the table. We want to see if this new ordered pair (8, -8) fits the pattern established by the other pairs.

Looking at the table, we can see that as x increases, y decreases. This makes sense because our function y = 8 - 2x has a negative slope (-2). This means that for every increase of 1 in x, y decreases by 2. Let's examine the changes in y as x increases in the table.

From x = -3 to x = -1, x increases by 2, and y decreases from 14 to 10, which is a decrease of 4. This confirms our slope of -2 (a change of -4 in y for every change of 2 in x). From x = -1 to x = 1, x increases by 2, and y decreases from 10 to 6, again a decrease of 4. From x = 1 to x = 4, x increases by 3, and y decreases from 6 to 0, a decrease of 6 (which still maintains the slope of -2). If we continue this pattern, when x goes from 4 to 8 (an increase of 4), we expect y to decrease by 8 (4 * -2 = -8). Starting from y = 0 when x = 4, a decrease of 8 gives us y = -8, which perfectly matches our calculated value.

This verification step is crucial because it helps us catch any potential errors. If our calculated y-value didn't fit the pattern, we'd know we made a mistake somewhere and need to double-check our work. Moreover, understanding the pattern reinforces the concept of linear functions and their consistent rate of change. By visually inspecting the table and comparing the differences in x and y values, we gain a deeper understanding of the function's behavior. In addition to verifying the pattern, we can also conceptually understand how the y-value changes as x increases. The consistent decrease in y for each increment in x provides a clear picture of the function's linearity. Let's wrap things up by summarizing what we've learned.

Conclusion: Mastering Functions and Ordered Pairs

Awesome! We've successfully found the value of y when x = 8 for the function y = 8 - 2x. We did it by substituting 8 for x in the equation and simplifying. We then verified our answer by checking if it fit the pattern established in the table of ordered pairs.

Key takeaways from this exercise:

• Functions define a relationship between input (x) and output (y).
• Ordered pairs (x, y) represent points on the graph of a function.
• Substitution is a powerful tool for finding unknown values in equations.
• Verifying your solution is crucial to ensure accuracy.

Understanding functions and ordered pairs is fundamental to algebra and many other areas of mathematics. You'll encounter these concepts again and again, so it's important to have a solid grasp of them. The process we used today – substituting and simplifying – is a skill you'll use throughout your math journey. Keep practicing, and you'll become a pro in no time!

Remember, guys, math isn't about memorizing formulas; it's about understanding the concepts and how they connect. By breaking down problems into smaller steps and thinking logically, you can tackle even the trickiest questions. So, keep exploring, keep questioning, and keep learning! This particular problem demonstrates the importance of understanding the function's rule and how to apply it. The ability to substitute values into an equation and solve for the unknown is a core skill in algebra. Furthermore, verifying the solution within the context of the given data reinforces the understanding of the function's behavior and the consistency of its rule. Next time you encounter a similar problem, remember the steps we took today: understand the function, substitute the given value, simplify the equation, and verify your solution. You've got this!