Evaluating $10-(2y-6)-y$ When Y=8 A Step-by-Step Guide

Hey guys! Today, we're diving into a fun little algebraic expression. We've got $10-(2y-6)-y$, and our mission, should we choose to accept it (and we totally do!), is to figure out what this bad boy equals when $y$ is 8. Sounds like a math adventure, right? Let's break it down step-by-step, making sure everyone's on board and having a good time. We'll go slow and steady, no need to rush. Think of it like baking a cake – you gotta follow the recipe to get the delicious results!

Step 1: Substitution

First things first, we need to substitute the value of $y$ into our expression. This basically means we're going to replace every instance of $y$ with the number 8. So, our expression $10-(2y-6)-y$ transforms into $10-(2(8)-6)-8$. See what we did there? We just swapped out the $y$ with an 8. Easy peasy, right? Substitution is a fundamental concept in algebra, and it's something you'll use a lot, so getting comfortable with it now is a great idea. It's like learning the alphabet before you can read – you gotta know the basics!

Now, let’s talk a bit more about why substitution is so important. In algebra, variables like $y$ are like placeholders. They can represent different numbers depending on the situation. When we're given a specific value for a variable, like $y=8$, we can plug that value into the expression to find its numerical value. This is super useful in solving equations and modeling real-world scenarios. Imagine you have a formula that calculates the cost of something based on the number of items you buy. The number of items would be a variable, and by substituting different values, you can find the cost for different quantities. Cool, huh?

Also, it's crucial to be meticulous when you're substituting. Make sure you're replacing every instance of the variable with its value. It's easy to miss one, especially in longer expressions, and that can throw off your entire calculation. Double-check your work! It's like proofreading a paper – a fresh pair of eyes (or a second look) can catch mistakes you might have missed the first time. And remember, parentheses are your friends! They help keep things organized and ensure you're following the correct order of operations, which we'll talk about next.

Step 2: Order of Operations (PEMDAS/BODMAS)

Okay, now that we've substituted, it's time to tackle the order of operations. This is where PEMDAS or BODMAS comes into play. You might have heard of it – it's an acronym that helps us remember the correct order to perform mathematical operations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Think of it as the rules of the road for math! If we don't follow these rules, we might end up with the wrong answer, and nobody wants that. It's like trying to build a house without a blueprint – it might look okay at first, but it's probably not going to be very stable.

Looking at our expression, $10-(2(8)-6)-8$, we see parentheses, so that's where we'll start. Inside the parentheses, we have $2(8)-6$. According to PEMDAS/BODMAS, we need to do the multiplication before the subtraction. So, $2(8)$ equals 16. Now our expression inside the parentheses becomes $16-6$. Subtracting 6 from 16 gives us 10. So, the stuff inside the parentheses simplifies to 10. Our entire expression now looks like $10-10-8$. We're getting there!

Let's pause for a moment and appreciate the power of PEMDAS/BODMAS. Without a consistent order of operations, math would be a chaotic mess! Imagine if everyone did calculations in whatever order they felt like – we'd get a bunch of different answers for the same problem. PEMDAS/BODMAS provides a universal standard, ensuring that everyone arrives at the same correct solution. It's like having a common language for math, allowing us to communicate and understand each other clearly. This is especially important in more complex calculations and scientific applications where accuracy is critical.

Furthermore, understanding the order of operations isn't just about getting the right answer in math problems. It's also about developing logical thinking and problem-solving skills. It teaches us to break down complex problems into smaller, more manageable steps, and to approach them in a systematic way. These skills are valuable in all areas of life, from cooking a meal to planning a project at work. So, mastering PEMDAS/BODMAS is an investment in your overall cognitive abilities!

Step 3: Simplifying from Left to Right

Alright, we've conquered the parentheses, and now we're left with $10-10-8$. Since we only have subtraction operations remaining, we'll work from left to right, just like reading a sentence. First up, we have $10-10$, which equals 0. So our expression simplifies to $0-8$. Almost there!

Now, subtracting 8 from 0 gives us -8. And that's it! We've reached the end of our mathematical journey. The value of the expression $10-(2y-6)-y$ when $y=8$ is -8. High five! We did it!

Working from left to right when dealing with addition and subtraction (or multiplication and division) is crucial for maintaining accuracy. It's another aspect of the order of operations that's often overlooked, but it can make a big difference in the final result. Think of it like a relay race – you have to pass the baton in the correct order to win. Similarly, you need to perform the operations in the correct order to arrive at the correct answer.

To illustrate this, let's consider what would happen if we didn't work from left to right. If we mistakenly subtracted 8 from 10 first, we'd get 2, and then subtracting that from 10 would give us 8. That's the opposite of our correct answer! This clearly shows why following the left-to-right rule is so important. It ensures that we're applying the operations in the way they were intended, and that we're not inadvertently changing the meaning of the expression.

Moreover, practicing this left-to-right approach helps build a strong foundation for more advanced mathematical concepts. As you progress in math, you'll encounter more complex expressions and equations, and having a solid understanding of the basic rules will be essential. It's like learning to dribble before you can play basketball – you need to master the fundamentals before you can excel at the game. So, keep practicing, and you'll become a math whiz in no time!

Step 4: The Grand Finale - The Answer!

So, to recap our exciting math adventure: We started with the expression $10-(2y-6)-y$, we substituted $y$ with 8, we carefully followed the order of operations (PEMDAS/BODMAS), and we simplified from left to right. And after all that hard work, we arrived at our final answer: -8. Woohoo! Give yourselves a pat on the back, guys! You've successfully navigated the world of algebraic expressions and come out victorious.

But wait, there's more! It's not just about getting the right answer; it's also about understanding the process. By breaking down the problem into smaller steps, we made it much easier to solve. This is a valuable skill that you can apply to all sorts of problems, not just math problems. It's like learning to cook a complicated dish – you don't try to do everything at once. You follow the recipe, step by step, and eventually, you have a delicious meal.

Remember, math is like a puzzle. Each problem is a new challenge, and each step you take is a piece of the puzzle. Sometimes the puzzle is easy, and sometimes it's a bit trickier, but with patience and perseverance, you can always solve it. And the feeling of satisfaction you get when you finally crack the code is totally worth it!

And that's a wrap, guys! We hope you enjoyed this math journey as much as we did. Keep practicing, keep exploring, and keep having fun with math. You're all math superstars in the making!

Conclusion

In conclusion, we've successfully evaluated the expression $10-(2y-6)-y$ when $y=8$. By following the order of operations and breaking down the problem step-by-step, we arrived at the solution of -8. Remember, math is a journey, not a destination. Keep practicing, keep learning, and keep having fun! You've got this!