Solving Algebraic Expressions Which Expression Is Equivalent To (x²-y²)-(6x²+4xy-y²)

Hey guys! Today, we're diving deep into the world of algebraic expressions, and we're going to tackle a question that might seem a bit intimidating at first glance. But trust me, with a little bit of algebraic magic, we'll break it down and find the solution together. Our mission is to figure out which expression is equivalent to the given expression: $(x^2-y2)-(6x^2+4xy-y2)$.

The Initial Algebraic Challenge: Understanding the Expression

Let's start by understanding what the question is really asking. We're given a seemingly complex expression, involving variables x and y, and a set of operations that we need to simplify. The key here is to remember our order of operations and to carefully combine like terms. Like terms are those that have the same variables raised to the same powers. For example, $x^2$ and $6x^2$ are like terms, but $x^2$ and $4xy$ are not, because even though they both involve x, the powers and other variables are different. This initial assessment is crucial because it guides our strategy. We need to distribute, combine, and simplify, ensuring we adhere to the fundamental rules of algebra. A slight misstep in this phase can lead to an incorrect final answer. So, focusing on the basics and double-checking each step will help us navigate the complexity and head toward the correct solution. This foundational understanding sets the stage for the subsequent algebraic manipulations.

The Distribution Phase: Removing Parentheses Like a Pro

The next step is to get rid of those pesky parentheses. Remember, when we have a negative sign in front of a parenthesis, we need to distribute that negative sign to every term inside. This is a crucial step because forgetting to distribute the negative sign is a common mistake that can lead to the wrong answer. So, let's distribute the negative sign in front of the second set of parentheses: $-(6x^2 + 4xy - y^2)$. When we distribute, we change the sign of each term inside the parentheses. So, $6x^2$ becomes $-6x^2$, $4xy$ becomes $-4xy$, and $-y^2$ becomes $+y^2$. Now, our expression looks like this: $x^2 - y^2 - 6x^2 - 4xy + y^2$. Distributing the negative sign correctly sets the foundation for combining like terms in the next phase, a critical process for simplifying the expression. Accuracy in distribution is paramount, as it directly influences the final outcome. Taking the time to meticulously apply this step prevents errors and keeps us on the right track.

Combining Like Terms: The Art of Algebraic Grouping

Now comes the fun part: combining like terms! This is where we group together the terms that have the same variables raised to the same powers. Let's identify our like terms. We have $x^2$ and $-6x^2$, which are like terms because they both have $x^2$. We also have $-y^2$ and $+y^2$, which are like terms because they both have $y^2$. And then we have $-4xy$, which is the only term with both x and y, so it's in a group of its own for now. Now, let's combine the like terms. $x^2 - 6x^2$ gives us $-5x^2$. $-y^2 + y^2$ cancels out and becomes 0. And $-4xy$ stays as it is. So, our simplified expression looks like this: $-5x^2 - 4xy$. Combining like terms is a fundamental algebraic skill, essential for simplifying expressions and solving equations. By accurately grouping and adding or subtracting coefficients, we reduce complexity and reveal the expression's underlying structure. This process not only simplifies the expression but also makes it easier to work with in further calculations or problem-solving scenarios.

The Final Solution: Spotting the Correct Answer

After all that algebraic maneuvering, we've arrived at our simplified expression: $-5x^2 - 4xy$. Now, we need to compare this to the answer choices provided. Looking at the options, we can see that option A, $-5x^2 - 4xy$, matches our simplified expression perfectly. So, that's our answer! We've successfully navigated through the expression, distributed the negative sign, combined like terms, and found the equivalent expression. This final step reinforces the importance of accurate algebraic manipulation and the satisfaction of arriving at the correct solution. The process highlights how each step, from initial understanding to the final comparison, plays a crucial role in solving algebraic problems. With practice and attention to detail, complex expressions can be simplified and matched with the correct answers efficiently.

Practice Makes Perfect: Honing Your Algebraic Skills

Algebra, like any skill, gets easier with practice. The more you work with expressions, distribute, and combine like terms, the more comfortable you'll become with the process. Don't be afraid to make mistakes – they're a natural part of learning. Just remember to learn from them and keep practicing. Try tackling similar problems, changing the coefficients or the signs, to really solidify your understanding. And if you ever get stuck, don't hesitate to ask for help. There are plenty of resources available online and in textbooks to help you master algebra. Remember, the key to success in algebra is consistent practice and a willingness to learn from your mistakes. By regularly engaging with problems, you'll build confidence and proficiency in manipulating algebraic expressions. Practice not only reinforces the concepts but also helps in recognizing patterns and shortcuts, making problem-solving more efficient and enjoyable.

Conclusion: Celebrating Our Algebraic Victory

So, there you have it, guys! We've successfully found the expression equivalent to $(x^2-y2)-(6x^2+4xy-y2)$. It's $-5x^2 - 4xy$, which corresponds to option A. Remember, algebra might seem intimidating at first, but with a step-by-step approach and a solid understanding of the basics, you can conquer any algebraic challenge. Keep practicing, keep exploring, and keep having fun with math! Each problem solved is a step forward in mastering algebraic skills. The ability to break down complex expressions into simpler forms is not only valuable in mathematics but also in various real-world applications. This success reinforces the power of perseverance and the effectiveness of systematic problem-solving. As we conclude, let's carry this confidence and approach future algebraic challenges with enthusiasm and determination.