Equivalent Expressions: A Step-by-Step Guide

Hey guys! Ever stumbled upon a math problem that looks like a jumbled mess of numbers and letters? Today, we're going to break down one of those problems and make it super easy to understand. We're diving into the world of algebraic expressions and figuring out which ones are actually the same, even if they look different. Our main goal is to figure out which expressions are equivalent to $-2y - 8 + 4y$. Sounds like fun, right? Let's get started!

Understanding the Initial Expression: $-2y - 8 + 4y$

First things first, let's really get to grips with what this expression, $-2y - 8 + 4y$, is all about. When you first look at it, it might seem a bit daunting, but trust me, it's simpler than it looks. The expression is made up of a few key parts: terms. In algebra, a term is a single number, a variable (like our 'y'), or numbers and variables multiplied together. In our case, we have three terms: $-2y$, $-8$, and $4y$. The $-2y$ term means -2 multiplied by our variable y. Variables, you know, are those letters that stand in for unknown numbers. Then we've got -8, which is just a plain old constant – a number that doesn't change. Lastly, we have $4y$, which means 4 multiplied by y. Now, the most important thing we need to remember here is the concept of like terms. Like terms are terms that have the same variable raised to the same power. In our expression, $-2y$ and $4y$ are like terms because they both have 'y' to the power of 1 (we usually don't write the power of 1, but it's there!). The -8 is a loner; it's a constant term and doesn't have a 'y' to hang out with. Simplifying expressions is all about combining these like terms to make the expression as neat and tidy as possible. So, what does combining like terms actually mean? It's just adding or subtracting the coefficients (the numbers in front of the variables) of the like terms. This is where the magic happens, and we start to see the hidden simplicity in the expression. This skill is super crucial, not just for this specific problem, but for loads of algebra problems you'll encounter. It's like having a secret decoder ring for math! So, let's roll up our sleeves and get ready to simplify.

Step-by-Step Simplification of $-2y - 8 + 4y$

Okay, let's dive right into simplifying the expression $-2y - 8 + 4y$. Remember, the name of the game here is combining like terms. We've already identified that $-2y$ and $4y$ are our like terms – they both have the variable 'y'. The -8 is just chilling by itself, a constant term, so we'll leave it alone for now. When we combine like terms, we're essentially just adding or subtracting their coefficients. The coefficient is the number that's multiplied by the variable. So, for $-2y$, the coefficient is -2, and for $4y$, it's 4. To combine them, we perform the operation -2 + 4. Think of it like this: if you're 2 steps behind and then you move 4 steps forward, where are you? You're 2 steps ahead! So, -2 + 4 equals 2. This means that when we combine $-2y$ and $4y$, we get $2y$. Now, let's bring back our constant term, the -8. Since it doesn't have any like terms to combine with, it just hangs out at the end of our expression. So, after combining like terms, our original expression $-2y - 8 + 4y$ simplifies beautifully to $2y - 8$. See? It's much cleaner and easier to look at now. This simplified form is super important because it's the key to figuring out which of the answer choices are equivalent to our original expression. We've essentially boiled down the expression to its core, and now we can compare it to the other options to see which ones match. This process of simplifying is a fundamental skill in algebra. It's like taking a messy room and organizing it so you can find everything easily. And in math, a simplified expression makes it much easier to solve equations, understand relationships, and avoid making mistakes. So, keep practicing this, and you'll become a simplification pro in no time!

Analyzing Option (A): $-2(y + 4) + 4y$

Alright, let's put on our detective hats and investigate option (A): $-2(y + 4) + 4y$. At first glance, it looks pretty different from our simplified expression, $2y - 8$, but don't let that fool you! We need to use a little algebraic magic to see if they're secretly the same. The first thing we notice in option (A) is the parentheses: (y + 4). This is a big clue that we need to use the distributive property. The distributive property is a fundamental rule in algebra that tells us how to multiply a number by a group of terms inside parentheses. It's like sharing candy with your friends – you have to give some to each of them! In this case, we're multiplying -2 by the entire group (y + 4). So, we need to multiply -2 by y and -2 by 4. When we multiply -2 by y, we get -2y. And when we multiply -2 by 4, we get -8. So, distributing the -2 across (y + 4) gives us -2y - 8. Now, let's bring down the rest of the expression, which is +4y. So, after distributing, option (A) now looks like this: $-2y - 8 + 4y$. Wait a minute… doesn't that look familiar? It's exactly the same as our original expression! But we know that our original expression simplifies to $2y - 8$. So, now we need to simplify this new version to see if it also becomes $2y - 8$. Just like before, we need to combine like terms. We have -2y and +4y, which are like terms because they both have the variable 'y'. When we combine them, we get 2y. And the -8 is still hanging out by itself. So, $-2y - 8 + 4y$ simplifies to $2y - 8$. Bingo! Option (A) simplifies to the same expression as our original, which means they are equivalent. This shows us the power of the distributive property and how it can help us transform expressions. Option (A) is definitely a match!

Evaluating Option (B): $4(-2 + y) - 2y$

Now, let's turn our attention to option (B): $4(-2 + y) - 2y$. Just like with option (A), we need to carefully unpack this expression and see if it's equivalent to our simplified form, $2y - 8$. The first thing that jumps out at us again is the parentheses: (-2 + y). This is another signal that the distributive property is going to come into play. Remember, the distributive property is our tool for multiplying a number by a group of terms inside parentheses. In this case, we're multiplying 4 by the entire group (-2 + y). So, we need to multiply 4 by -2 and 4 by y. When we multiply 4 by -2, we get -8. And when we multiply 4 by y, we get 4y. So, distributing the 4 across (-2 + y) gives us -8 + 4y. Now, let's bring down the rest of the expression, which is -2y. So, after distributing, option (B) looks like this: -8 + 4y - 2y. The next step, just like before, is to simplify by combining like terms. In this expression, we have 4y and -2y as our like terms. They both have the variable 'y', so we can combine their coefficients. When we combine 4y and -2y, we're essentially doing 4 - 2, which equals 2. So, 4y - 2y simplifies to 2y. Now, let's bring down the constant term, which is -8. It doesn't have any like terms to combine with, so it stays as it is. Putting it all together, -8 + 4y - 2y simplifies to -8 + 2y. Hmmm… this looks pretty close to our simplified expression, $2y - 8$, but the order is a bit different. But guess what? Addition is commutative, which means we can change the order of the terms without changing the value of the expression. So, -8 + 2y is exactly the same as $2y - 8$! That means option (B) is also equivalent to our original expression. We've found another match! It's pretty cool how the distributive property and combining like terms can reveal hidden connections between expressions, right?

Final Answer and Wrap-up

Okay, guys, let's bring it all together and nail down the final answer! We started with the expression $-2y - 8 + 4y$ and simplified it to $2y - 8$. Then, we put on our detective hats and analyzed two options:

• Option (A): $-2(y + 4) + 4y$. After using the distributive property and combining like terms, we found that it also simplified to $2y - 8$. So, option (A) is definitely equivalent.
• Option (B): $4(-2 + y) - 2y$. Again, we used the distributive property and combined like terms, and guess what? It also simplified to $2y - 8$! So, option (B) is also a match.

So, the final answer is that both options (A) and (B) are equivalent to our original expression, $-2y - 8 + 4y$. You nailed it! Understanding how to simplify expressions and use the distributive property is a super important skill in algebra. It's like having a secret weapon that lets you tackle even the trickiest-looking problems. By breaking down expressions into smaller parts, combining like terms, and using properties like the distributive property, you can reveal the hidden simplicity beneath the surface. Remember, practice makes perfect! The more you work with these concepts, the easier they'll become. And the more comfortable you are with algebra, the more confident you'll feel tackling any math challenge that comes your way. So, keep up the great work, and keep exploring the amazing world of math!