Solving For Y When X Is 8 In The Equation Y = (x^2 / 4) - 2
Hey there, math enthusiasts! Today, we're diving into a fun little algebra problem. We're given an equation and a value for one variable, and our mission is to find the value of the other variable. So, let's jump right in and break it down step-by-step. We aim to provide you with a comprehensive explanation that not only solves the problem but also enhances your understanding of algebraic concepts. Get ready to sharpen your math skills and tackle this challenge head-on!
Understanding the Problem
First, let's understand the problem clearly. We have an equation: y = (x² ÷ 4) - 2. This equation tells us how y is related to x. We're also given that x = 8. Our goal is to find the value of y when x is 8. This is a classic substitution problem, a fundamental concept in algebra. Understanding the relationships between variables and how to manipulate equations is crucial for solving various mathematical problems. Now, let's dive deeper into how we can solve this step-by-step.
Breaking Down the Equation
The equation y = (x² ÷ 4) - 2 might look a bit intimidating at first, but let's break it down. The left side, y, is what we want to find. The right side is an expression involving x. This expression consists of a few operations: squaring x, dividing by 4, and subtracting 2. Order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which we perform these operations. This ensures we arrive at the correct answer. Understanding and applying PEMDAS is essential in algebraic manipulations and will help you avoid common mistakes. Now, let's see how this applies to our specific problem.
Solving for y
Now, let's put on our problem-solving hats and find the value of y. Remember, we know that x = 8. The key here is substitution – replacing the variable x with its given value in the equation. This transforms the equation from one involving two variables to one involving just y, which we can then solve directly. Substitution is a powerful tool in algebra and is frequently used to solve systems of equations and evaluate expressions. Now, let's perform the substitution and see what we get!
Step 1: Substitute x = 8
Our first step is to substitute x = 8 into the equation y = (x² ÷ 4) - 2. This means we replace every instance of x in the equation with the number 8. So, the equation becomes: y = (8² ÷ 4) - 2. By doing this, we've successfully eliminated the variable x and now have an equation that only involves y. This is a crucial step because it allows us to isolate y and find its value. The careful substitution is vital for ensuring accuracy in the subsequent calculations.
Step 2: Calculate 8²
Next up, we need to calculate 8². Remember, 8² (8 squared) means 8 multiplied by itself: 8 * 8. So, 8² = 64. This is an example of an exponent, a mathematical notation that indicates repeated multiplication. Understanding exponents and how to calculate them is fundamental to algebra and many other areas of mathematics. Now, let's substitute this result back into our equation. Our equation now looks like this: y = (64 ÷ 4) - 2.
Step 3: Divide 64 by 4
Now, let's tackle the division. We need to divide 64 by 4. Performing this division, we find that 64 ÷ 4 = 16. Division is one of the basic arithmetic operations, and it's important to get it right! With this calculation, our equation simplifies further. We now have: y = 16 - 2. We're getting closer to finding the value of y. Just one more step to go!
Step 4: Subtract 2 from 16
Finally, we subtract 2 from 16. This is a straightforward subtraction: 16 - 2 = 14. And that's it! We've found the value of y. This final subtraction gives us the solution to the problem. So, we can confidently say that when x = 8, the value of y is 14.
The Answer
So, after all our calculations, we've arrived at the answer. When x = 8, the value of y in the equation y = (x² ÷ 4) - 2 is 14. Therefore, the correct answer is A. 14. We've successfully navigated through the equation, performed the necessary operations, and found the solution. Great job, guys! You've tackled an algebraic problem with confidence and precision.
Key Takeaways
Let's recap the key takeaways from this problem. First, we learned the importance of substitution in solving equations. Substituting the given value of x allowed us to transform the equation into a solvable form. Second, we reinforced the significance of the order of operations (PEMDAS) in ensuring accurate calculations. Following the correct order is crucial to avoid errors. Finally, we practiced basic arithmetic operations such as squaring, division, and subtraction. Mastering these fundamental skills is essential for success in algebra and beyond. By understanding these concepts, you'll be well-equipped to tackle similar problems in the future!
Practice Makes Perfect
Guys, remember, the more you practice, the better you'll get at solving these types of problems. Try changing the value of x and recalculating y. You could also modify the equation slightly, for example, by adding another term or changing the division factor. The key is to keep challenging yourself and building your skills. Math is like a muscle – the more you exercise it, the stronger it becomes. So, keep practicing, keep learning, and keep having fun with math! You've got this!
Real-World Applications
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