Simplifying Expressions: Unveiling Equivalencies

Hey math enthusiasts! Today, we're diving into the world of algebraic expressions and uncovering the secrets of simplification. Our main objective? To determine which expressions are equivalent to $-2(2h + 10) + 4h$. It might seem like a maze at first, but trust me, with a few simple steps, we'll crack the code and find the equivalent expressions. Let's break it down, step by step, and make sure everyone understands the process. This isn't just about getting the right answer; it's about understanding why that answer is correct and mastering the art of simplifying expressions. Get ready to flex those math muscles and learn some cool tricks!

Understanding the Expression: $-2(2h + 10) + 4h$

First off, let's take a good look at the expression we're dealing with: $-2(2h + 10) + 4h$. At its heart, this is a combination of multiplication and addition, with a variable, 'h', thrown into the mix. Our main goal is to simplify it as much as possible, which means rewriting it in a way that's easier to understand and use. Remember the order of operations (PEMDAS/BODMAS)? We'll be using it to guide our steps, ensuring we don't make any errors along the way. In this expression, we have parentheses, multiplication, and addition. The key to simplifying this type of expression lies in applying the distributive property, which is like a magic key that unlocks the parentheses, and then combining like terms. It's like a math puzzle, and we're the detectives, trying to solve it. Let's start with the distributive property!

The Power of the Distributive Property

The distributive property is a fundamental concept in algebra. It tells us how to handle expressions where a number is multiplied by a sum or difference inside parentheses. In our expression, we have $-2$ multiplying the quantity $(2h + 10)$. The distributive property tells us that we need to multiply the $-2$ by each term inside the parentheses. So, we'll multiply $-2$ by $2h$ and then $-2$ by $10$. It's a straightforward process, but it's crucial to get it right. Multiplying $-2$ by $2h$ gives us $-4h$, and multiplying $-2$ by $10$ gives us $-20$. This step effectively removes the parentheses, making the expression easier to work with. Remember to pay close attention to the signs – a negative times a positive results in a negative. The distributive property is a powerful tool, and it's essential for simplifying many algebraic expressions. It’s like the secret handshake to opening up these kinds of math problems. So, what do we have after applying the distributive property? We have $-4h - 20$. Now what?

Combining Like Terms

After applying the distributive property, our expression now looks like this: $-4h - 20 + 4h$. Our next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, we have two terms with the variable 'h': $-4h$ and $+4h$. We can combine these terms by adding their coefficients (the numbers in front of the variables). In this case, we have $-4 + 4$, which equals 0. So, when we combine $-4h$ and $+4h$, they cancel each other out, leaving us with 0h, or simply 0. What's left? Only the constant term -20! The ability to identify and combine like terms is a cornerstone of algebra, allowing us to simplify expressions and solve equations. It’s like sorting your laundry – you group similar items together to make the process easier. So, after combining like terms, our simplified expression is simply -20.

Evaluating the Answer Choices

Now that we've simplified our original expression, let's look at the answer choices and see which one matches our simplified form. We've done the hard work, so now it's time to see if we were successful! This is where we confirm our work and ensure that our understanding is on point. Remember, in math, it's always smart to double-check your work, and this is a great way to do that.

Analyzing the Options

We need to compare our final, simplified expression (-20) with the provided answer choices. Here's a look at the options:

• A. $-20 - 4h + 4h$: Let's simplify this expression to see if it matches ours. Combining the like terms $-4h$ and $+4h$, we get 0. This leaves us with just -20. This option appears to be equivalent.
• B. $-20$: This is exactly what we got after simplifying our original expression. This option is certainly equivalent.
• C. None of the above: Given our calculations, we know that this option is incorrect, as we have identified equivalent expressions.

The Correct Match

Based on our calculations, both options A and B are equivalent to the original expression. Option A simplifies to -20 and option B is already -20. This means the correct answer is not just one option, but multiple. Therefore, you must select the appropriate answers, as it is based on your understanding of the question.

Conclusion: The Simplified Result

So, there you have it! We started with a complex-looking expression, $-2(2h + 10) + 4h$, and, by using the distributive property and combining like terms, we simplified it to $-20$. Not only did we find the simplified form, but we also identified which of the answer choices were equivalent. The goal of simplifying expressions is to make them easier to understand and work with, and in this case, we've shown how the original expression can be reduced to a much simpler form. Great job, guys! Remember, practice makes perfect, so keep working on these types of problems, and you'll become a pro in no time! Remember, understanding the why behind the math is just as important as getting the right answer. Keep up the excellent work, and always remember to double-check those calculations!