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

by ADMIN 39 views

Hey guys! Today, we're diving into a super common type of math problem that you'll likely encounter in algebra: finding the value of y in a function when you're given a specific value for x. In this case, we have the function y = 8 - 2x, and we want to figure out what y is when x is 8. Let's break it down step by step so you can master this skill!

Understanding the Function y = 8 - 2x

Before we jump into plugging in numbers, let's make sure we understand what this function, y = 8 - 2x, actually means. Think of it as a little machine. You feed it a number (x), it does some calculations, and then it spits out another number (y). The equation tells us exactly what calculations the machine performs:

  1. First, it multiplies the input (x) by 2.
  2. Then, it subtracts that result from 8.

The result of this calculation is the output, which is the value of y. So, y depends on the value of x. That's why we call y the dependent variable and x the independent variable. This understanding of functions is fundamental, guys, so make sure you're solid on this concept.

Ordered Pairs and Tables

The problem gives us a table of ordered pairs. An ordered pair is simply a pair of numbers, (x, y), that represents a point on a graph or a solution to the equation. The table shows us several ordered pairs that satisfy the equation y = 8 - 2x. For instance, the table shows that when x = -3, y = 14. This means that the ordered pair (-3, 14) is a solution to the equation. Looking at the table helps us see the relationship between x and y more clearly.

The Problem: Finding y When x = 8

Our mission, should we choose to accept it (and we do!), is to find the value of y when x is 8. The table actually gives us a hint! We see the row where x = 8, but the y value is marked with a question mark. That's our target! We need to use the function y = 8 - 2x to calculate the value of y when x is 8. This is a classic example of function evaluation.

Step-by-Step Solution: Plugging in the Value of x

Okay, let's get to the math! Here's how we solve this problem:

  1. Write down the function: This helps keep us organized. We have y = 8 - 2x.
  2. Substitute the value of x: We're given that x = 8, so we replace x in the equation with 8. This gives us y = 8 - 2(8).
  3. Perform the multiplication: According to the order of operations (PEMDAS/BODMAS), we do multiplication before subtraction. So, we calculate 2(8) which equals 16. Now our equation looks like this: y = 8 - 16.
  4. Perform the subtraction: Finally, we subtract 16 from 8. 8 - 16 = -8. This is where paying attention to negative numbers is crucial!
  5. Write the answer: We've found that when x = 8, y = -8. So, the value of y is -8.

The Solution: y = -8

So, the answer to the question "What is the value of y when x = 8?" is y = -8. We successfully evaluated the function! We can now confidently fill in the question mark in the table with -8.

Why This Matters: Real-World Applications of Functions

You might be thinking, "Okay, that's cool, but why do we even care about this?" Well, guys, functions are everywhere in the real world! They're used to model relationships between different quantities. Here are a few examples:

  • Physics: The distance an object travels can be a function of time and speed.
  • Economics: The profit a company makes can be a function of the number of products sold.
  • Computer Science: Functions are the building blocks of computer programs.

Understanding how to work with functions, like we did in this problem, is a fundamental skill that will help you in many different fields. Mastering this now sets you up for success later.

Practice Makes Perfect: More Examples and Exercises

The best way to get really good at working with functions is to practice! Let's look at a couple of more examples:

Example 1:

What is the value of y in the function y = 3x + 5 when x = 2?

  1. Write down the function: y = 3x + 5
  2. Substitute x = 2: y = 3(2) + 5
  3. Multiply: y = 6 + 5
  4. Add: y = 11

So, when x = 2, y = 11.

Example 2:

What is the value of y in the function y = x² - 4 when x = -1?

  1. Write down the function: y = x² - 4
  2. Substitute x = -1: y = (-1)² - 4
  3. Calculate the exponent: Remember that (-1)² = (-1) * (-1) = 1. So, y = 1 - 4
  4. Subtract: y = -3

So, when x = -1, y = -3.

Practice Exercises

Now it's your turn! Try these exercises on your own:

  1. What is the value of y in the function y = -x + 7 when x = 4?
  2. What is the value of y in the function y = 2x² - 1 when x = 3?
  3. What is the value of y in the function y = (1/2)x - 6 when x = 10?

Work through these problems step-by-step, and you'll become a function master in no time! Don't be afraid to make mistakes; that's how we learn.

Common Mistakes to Avoid

Even though the process is pretty straightforward, there are a few common mistakes that students make when evaluating functions. Let's go over them so you can avoid them:

  • Order of Operations: Always, always, always follow the order of operations (PEMDAS/BODMAS)! Multiplication and division come before addition and subtraction. Exponents come before multiplication and division.
  • Negative Signs: Be extra careful when dealing with negative numbers. Make sure you understand how to multiply and subtract with negatives. Remember that a negative times a negative is a positive!
  • Substitution Errors: Double-check that you've substituted the value of x correctly into the equation. It's easy to make a small mistake here, especially if the equation is long or has many terms.

Being aware of these common pitfalls will help you solve function problems accurately and efficiently.

Conclusion: You've Got This!

So, guys, we've successfully found the value of y in the function y = 8 - 2x when x = 8. We've also discussed why functions are important and looked at some more examples and practice exercises. Remember the key steps: write down the function, substitute the value of x, and follow the order of operations. With a little practice, you'll become a pro at evaluating functions! Keep practicing, and you'll build a strong foundation in algebra.

If you have any questions, don't hesitate to ask! Keep up the great work, and I'll see you in the next lesson!