Simplifying Expressions: A Step-by-Step Guide

Hey math enthusiasts! Ever feel like algebraic expressions are a bit of a maze? Fear not, because today we're going to break down how to simplify them step by step! We'll tackle the expression $(2y - x) - 2(y - 2z) - 4(x + z)$ and see which of the multiple-choice options is equivalent. It's all about understanding the order of operations and being meticulous with your signs. Get ready to flex those math muscles! This guide will provide a clear, concise, and easy-to-follow explanation. We'll explore each step with explanations, ensuring you not only find the correct answer but also gain a solid understanding of the underlying principles. Let's dive in and demystify this problem together!

Understanding the Problem: The Core Concepts

Alright, before we get our hands dirty with the calculations, let's make sure we're all on the same page regarding the core concepts. What does it even mean to simplify an expression? Basically, it means making it as concise as possible while maintaining its original value. We're aiming to rewrite $(2y - x) - 2(y - 2z) - 4(x + z)$ in a form that is equivalent but has fewer terms and simpler coefficients. The key mathematical concepts at play here are the distributive property and combining like terms. The distributive property is what allows us to get rid of those pesky parentheses. It states that $a(b + c) = ab + ac$. For example, $2(3 + 4)$ is equal to $2 * 3 + 2 * 4$. We'll use this to deal with the $-2(y - 2z)$ and $-4(x + z)$ parts of our expression. Then, we need to know how to combine like terms. Like terms are terms that have the same variable raised to the same power. For instance, $3x$ and $-5x$ are like terms. We can add or subtract these terms by adding or subtracting their coefficients: $3x - 5x = -2x$. So, combining like terms involves adding or subtracting the coefficients of terms that share the same variable and exponent. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is also critical here. It dictates the sequence in which we perform the mathematical operations. We'll start by addressing the parentheses using the distributive property. Only then will we combine like terms. With a solid understanding of these concepts, simplifying any expression becomes much more manageable! We'll put these principles into action in the next section. Are you ready to get started, guys?

Step-by-Step Simplification: Unpacking the Expression

Now, let's roll up our sleeves and systematically simplify the expression $(2y - x) - 2(y - 2z) - 4(x + z)$. We'll break it down into manageable steps, ensuring clarity and accuracy. First, we have to address the parentheses using the distributive property. Starting with the term $-2(y - 2z)$, we multiply $-2$ by each term inside the parentheses:

$-2 * y = -2y$ $-2 * -2z = +4z$

So, $-2(y - 2z)$ becomes $-2y + 4z$. Now, let's tackle the term $-4(x + z)$. We distribute the $-4$ across the terms within the parentheses:

$-4 * x = -4x$ $-4 * z = -4z$

Thus, $-4(x + z)$ transforms into $-4x - 4z$. Now we can rewrite the entire expression, substituting the simplified forms of the terms with parentheses:

$(2y - x) - 2(y - 2z) - 4(x + z)$ becomes

$2y - x - 2y + 4z - 4x - 4z$

Great! We've successfully eliminated the parentheses. Next up, we will combine like terms to streamline the expression further. We have terms with $y$, terms with $x$, and terms with $z$. We can gather similar terms, which will make the process easier.

Combining Like Terms: The Final Touch

Alright, now that we've broken down the expression and applied the distributive property, the final step involves combining like terms. This will allow us to rewrite the expression in its simplest form. Let's start with the $y$ terms. We have $2y$ and $-2y$. When we combine them, we get:

$2y - 2y = 0$

So, the $y$ terms cancel each other out. Next, let's look at the $x$ terms. We have $-x$ and $-4x$. Combining them, we get:

$-x - 4x = -5x$

Finally, let's address the $z$ terms. We have $4z$ and $-4z$. When we combine them:

$4z - 4z = 0$

So, the $z$ terms also cancel each other out. Putting everything together, the simplified expression is:

$0 - 5x + 0$

Which simplifies to

$-5x$

And there you have it, folks! The simplified expression is $-5x$. This is a perfect example of how combining the distributive property and combining like terms leads to the solution. Always take your time and break down the problem step by step. This method will help you improve your math skills and make solving similar problems a breeze. Remember, practice makes perfect. Keep up the great work!

Matching the Answer: Choosing the Correct Option

We did it! We successfully simplified the expression $(2y - x) - 2(y - 2z) - 4(x + z)$ and found that it simplifies to $-5x$. Now, let's see which of the multiple-choice options matches our answer. The options were:

A) $-3x$ B) $-5x$ C) $-3x + 8z$ D) $-5x - 8z$

We found that the simplified expression is $-5x$, which perfectly matches option B) $-5x$. Therefore, the correct answer is B. Easy peasy, right? Always double-check your work to ensure you haven't made any mistakes during the simplification process. Compare the final answer with the given options, and choose the matching one. We can confirm our result by working backwards. If we substitute a value for x (let's say x = 1) into the original and simplified equations, we can see if they are the same.

Original equation: $(2y - x) - 2(y - 2z) - 4(x + z)$ Simplified equation: $-5x$

Let's assume y = 1 and z = 1 for the first equation:

$(2(1) - 1) - 2(1 - 2(1)) - 4(1 + 1)$ becomes $(2 - 1) - 2(1 - 2) - 4(2)$ $1 - 2(-1) - 8$ $1 + 2 - 8 = -5$

For the simplified equation $-5x$, we said $x=1$ so $-5(1) = -5$

Since both equations give -5, we know that our solution is correct. Congratulations! You've successfully simplified the expression and identified the correct answer. You now have a stronger grasp of how to deal with similar problems in the future. Always take the time to check your steps to avoid any mistakes. Great job!

Tips and Tricks: Mastering Expression Simplification

Want to become a simplification superstar? Here are some tips and tricks to help you master the art of simplifying expressions. Practice, practice, practice! The more you work through different problems, the more comfortable you'll become with the concepts and steps involved. Start with simpler problems and gradually move to more complex ones. Focus on understanding the fundamentals: the distributive property, combining like terms, and the order of operations. Once you truly understand these, you will solve almost any expression. Pay close attention to signs. A negative sign can make a big difference, so be extra careful when distributing and combining terms. Double-check your work at every step. This will help you catch any errors before they snowball into a bigger problem. Don't be afraid to break down problems into smaller steps. Writing out each step can help you avoid making mistakes and keep track of your progress. Use different examples. Try different expressions and experiment with different approaches to reinforce your understanding. Always be organized. Keep your work neat and well-organized to avoid confusion and make it easier to identify any potential errors. Consider using visual aids. Drawing diagrams or using color-coding can sometimes help you visualize the problem and keep track of the different terms. If you get stuck, don't give up! Look back at your notes, review the concepts, or ask for help from a teacher, friend, or online resource. Always remember to stay positive. Believe in your ability to learn and improve. With consistent effort and a positive attitude, you will get better at simplifying expressions. These tips and tricks will help you improve your skills and solve any similar problem. Good luck, and keep practicing!

Conclusion: Your Simplification Journey

Alright, guys, we've reached the end of our simplification adventure! We started with a complex expression and, step by step, broke it down to its simplest form. We explored the distributive property, combined like terms, and, most importantly, gained a deeper understanding of how to simplify algebraic expressions. Remember, the key to success in simplifying expressions, or any math problem, is practice and patience. Don't be discouraged if it doesn't come easily at first. Keep practicing, reviewing the concepts, and asking questions when you get stuck. The more you work with these types of problems, the more confident and proficient you will become. Always double-check your work to ensure accuracy. Identify the core concepts, and focus on the order of operations. Embrace the challenge and enjoy the process of learning. Keep up the excellent work, and you'll soon be tackling even the most complicated expressions with ease! Keep practicing and expanding your knowledge, and you'll be well on your way to mathematical success. You got this!