Simplifying 3(x-y)+2(-2x+6y) A Step-by-Step Guide

Hey guys! Let's dive into simplifying this algebraic expression: $3(x-y) + 2(-2x + 6y)$. It might look a bit intimidating at first, but don't worry, we'll break it down step by step. Algebraic expressions are the building blocks of algebra, and mastering them is crucial for tackling more complex math problems. In this article, we’ll not only simplify the given expression but also discuss the underlying principles and techniques that you can apply to various similar problems. Simplifying expressions involves combining like terms and using the distributive property to eliminate parentheses. Understanding these concepts thoroughly will enhance your algebraic skills and confidence.

Understanding the Basics of Algebraic Expressions

Before we jump into the solution, let’s quickly recap some foundational concepts. Algebraic expressions are combinations of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Variables are symbols (usually letters like x and y) that represent unknown values, while constants are fixed numerical values. For instance, in the expression $3(x-y) + 2(-2x + 6y)$, x and y are variables, and 3 and 2 are constants. Understanding the different components of algebraic expressions is the first step toward simplification. Coefficients are the numbers that multiply the variables, so in our expression, 3 and 2 are coefficients outside the parentheses, and -2 and 6 are coefficients inside the second set of parentheses. Terms are the individual parts of an expression separated by addition or subtraction. For example, $3(x-y)$ and $2(-2x + 6y)$ are two main terms in the given expression. To simplify algebraic expressions effectively, it’s essential to recognize and understand these elements. This foundational knowledge allows us to apply the appropriate rules and techniques, such as the distributive property and combining like terms, which we will explore in detail as we simplify the expression.

The Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms inside parentheses. It states that a(b + c) = ab + ac. This property is crucial for eliminating parentheses and simplifying expressions. In our case, we have $3(x-y)$ and $2(-2x + 6y)$, which both require the use of the distributive property. Let's break it down further: when we distribute 3 across (x - y), we multiply 3 by both x and -y, resulting in $3x - 3y$. Similarly, when we distribute 2 across (-2x + 6y), we multiply 2 by both -2x and 6y, which gives us $-4x + 12y$. The distributive property is not just a mechanical rule; it’s a logical extension of how multiplication and addition work together. By understanding the distributive property, we can efficiently expand expressions and prepare them for further simplification. The ability to correctly apply the distributive property is a key skill in algebra, and it's essential for simplifying a wide variety of expressions, including the one we are focusing on in this article. Mastering this property makes algebraic manipulations much more straightforward and reduces the likelihood of errors.

Combining Like Terms

Combining like terms is another essential technique in simplifying algebraic expressions. Like terms are terms that have the same variables raised to the same powers. For example, $3x$ and $-4x$ are like terms because they both have the variable x raised to the power of 1. Similarly, $-3y$ and $12y$ are like terms because they both have the variable y raised to the power of 1. Constants, such as 5 and -2, are also considered like terms because they don't have any variables. To combine like terms, we simply add or subtract their coefficients while keeping the variable part the same. For instance, to combine $3x$ and $-4x$, we add their coefficients (3 and -4), resulting in $-1x$, which is usually written as $-x$. Similarly, to combine $-3y$ and $12y$, we add their coefficients (-3 and 12), resulting in $9y$. Understanding how to identify and combine like terms is crucial for reducing an algebraic expression to its simplest form. This process not only makes the expression more concise but also easier to work with in further algebraic manipulations or when solving equations. The ability to accurately combine like terms is a foundational skill that significantly enhances your algebraic proficiency.

Step-by-Step Solution

Okay, let's tackle the expression $3(x-y) + 2(-2x + 6y)$ step by step. Remember, our goal is to simplify it by removing parentheses and combining like terms. This methodical approach will help ensure accuracy and clarity in our solution. Let's get started!

Step 1: Apply the Distributive Property

First, we'll use the distributive property to eliminate the parentheses. This involves multiplying the terms outside the parentheses by each term inside. For the first part, $3(x-y)$, we multiply 3 by both x and -y: $3 * x = 3x$ $3 * -y = -3y$ So, $3(x-y)$ becomes $3x - 3y$. Now, let’s apply the distributive property to the second part, $2(-2x + 6y)$. We multiply 2 by both $-2x$ and $6y$: $2 * -2x = -4x$ $2 * 6y = 12y$ Thus, $2(-2x + 6y)$ becomes $-4x + 12y$. Now we can rewrite the original expression as the sum of these simplified parts: $3x - 3y + (-4x + 12y)$. This step is crucial because it transforms the expression into a form where we can easily identify and combine like terms. The distributive property is a powerful tool, and mastering it is essential for simplifying more complex algebraic expressions. By breaking down the problem into smaller, manageable steps, we make the simplification process much clearer and less prone to errors.

Step 2: Identify Like Terms

Next, we need to identify the like terms in our expanded expression, which is $3x - 3y - 4x + 12y$. Remember, like terms have the same variable raised to the same power. In this expression, we have two terms with the variable x: $3x$ and $-4x$. We also have two terms with the variable y: $-3y$ and $12y$. Identifying like terms is a crucial step because it allows us to combine them and simplify the expression further. By grouping the like terms together mentally (or even visually by underlining or circling them), we can more easily see which terms can be combined. This step sets the stage for the final simplification by organizing the terms in a way that makes the next step, combining the terms, straightforward. Proper identification of like terms ensures that we are only combining terms that can be combined, maintaining the integrity of the expression and leading to the correct simplified form.

Step 3: Combine Like Terms

Now, let’s combine the like terms we identified in the previous step. We have the x terms: $3x$ and $-4x$. To combine them, we add their coefficients: $3 + (-4) = -1$. So, $3x - 4x$ simplifies to $-1x$, which we usually write as $-x$. Next, we combine the y terms: $-3y$ and $12y$. We add their coefficients: $-3 + 12 = 9$. Therefore, $-3y + 12y$ simplifies to $9y$. Putting it all together, our simplified expression is $-x + 9y$. This step is where the expression is reduced to its simplest form by adding or subtracting the coefficients of like terms. By accurately combining like terms, we not only make the expression more concise but also ensure that it is easier to work with in subsequent mathematical operations or problem-solving scenarios. This final combination is a testament to the power of algebraic simplification, transforming a potentially complex-looking expression into a much more manageable and understandable form.

Final Answer

So, after applying the distributive property and combining like terms, the simplified form of the expression $3(x-y) + 2(-2x + 6y)$ is $-x + 9y$. This is the most concise form of the expression, and it's equivalent to the original expression for all values of x and y. You did it! Simplifying algebraic expressions might seem tricky at first, but with practice, it becomes second nature. Remember the key steps: distribute, identify like terms, and combine. These steps are the cornerstone of algebraic simplification, and mastering them will significantly boost your confidence in handling algebraic problems. Keep practicing, and you'll become an algebra pro in no time!

Practice Problems

To solidify your understanding, let's try a couple of practice problems. Working through these examples will help you apply the techniques we've discussed and identify any areas where you might need further clarification. Practice is essential for mastering algebraic simplification, and these problems will give you valuable experience. Each problem offers a unique opportunity to reinforce your skills and build your confidence. Remember to follow the same steps we used earlier: apply the distributive property, identify like terms, and combine them. Let's dive in and get some more practice!

Practice Problem 1

Simplify: $4(2a + b) - 3(a - 2b)$

Practice Problem 2

Simplify: $-2(3x - 4y) + 5(x + y)$

Working through these practice problems will not only enhance your algebraic skills but also give you a sense of accomplishment as you see yourself successfully applying the simplification techniques. Remember, algebra is like any other skill – the more you practice, the better you become. So, grab a pencil and paper, and let's tackle these problems!

Conclusion

Alright, guys, we've covered a lot in this article! We started with the expression $3(x-y) + 2(-2x + 6y)$ and walked through the entire process of simplifying it. We revisited the distributive property and the importance of combining like terms. By breaking down the problem into manageable steps, we were able to transform the expression into its simplest form: $-x + 9y$. This process illustrates the fundamental principles of algebraic simplification, which are essential for tackling more complex mathematical problems. Remember, the key to mastering algebra is understanding the underlying concepts and practicing consistently. By applying the techniques we've discussed, you can confidently simplify a wide range of algebraic expressions. Keep up the great work, and you'll be amazed at how your algebraic skills develop over time! Algebraic simplification is not just a skill; it's a powerful tool that will serve you well in various areas of mathematics and beyond. So, embrace the challenge, keep practicing, and enjoy the journey of learning algebra!