Simplifying The Expression $12(1/6 X + 1/3 - 2/3 X - 2/3)$ A Step-by-Step Guide

In the realm of algebra, simplifying expressions is a fundamental skill. It allows us to manipulate equations and formulas into more manageable forms, making them easier to understand and solve. This article delves into the simplification of the expression $12(\frac{1}{6}x + \frac{1}{3} - \frac{2}{3}x - \frac{2}{3})$, providing a step-by-step guide and exploring the underlying mathematical principles. This detailed explanation will benefit students learning algebra and anyone looking to refresh their understanding of algebraic simplification. We'll break down each step, ensuring clarity and offering insights into the logic behind the process. By the end of this guide, you'll be well-equipped to tackle similar algebraic challenges with confidence. This process not only simplifies the expression but also enhances your overall algebraic problem-solving skills. Let's embark on this journey of algebraic simplification together, transforming a seemingly complex expression into its simplest form. Throughout this discussion, we aim to provide a clear and concise explanation, making the process accessible to learners of all levels. Whether you're a student preparing for an exam or simply curious about algebra, this guide will serve as a valuable resource.

Understanding the Expression: A Step-by-Step Breakdown

The expression we aim to simplify is $12(\frac{1}{6}x + \frac{1}{3} - \frac{2}{3}x - \frac{2}{3})$. This algebraic expression involves variables, fractions, and parentheses, which can seem daunting at first glance. However, by systematically applying the rules of algebra, we can break it down into simpler components. The first key step is to understand the order of operations, often remembered by the acronym PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which we perform operations to ensure we arrive at the correct answer. We will start by focusing on the terms inside the parentheses. Combining like terms within the parentheses is crucial for simplifying the expression. Like terms are those that contain the same variable raised to the same power. In our case, we have terms with 'x' and constant terms. By grouping and combining these, we can reduce the complexity of the expression. This step-by-step approach not only simplifies the expression but also reinforces the fundamental principles of algebraic manipulation. Understanding these principles is essential for tackling more advanced algebraic problems in the future. As we progress through the simplification process, we'll highlight the specific rules and techniques applied, ensuring a comprehensive understanding of each step. This thorough breakdown aims to empower you with the skills and knowledge necessary to confidently simplify similar expressions on your own. Remember, practice is key to mastering algebra, so working through examples like this one is an invaluable learning experience.

Step 1: Combining Like Terms Inside the Parentheses

In this crucial step, we focus on simplifying the expression within the parentheses: $(\frac{1}{6}x + \frac{1}{3} - \frac{2}{3}x - \frac{2}{3})$. The key here is to identify and 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 'x': $\frac{1}{6}x$ and $-\frac{2}{3}x$. We also have two constant terms: $\frac{1}{3}$ and $-\frac{2}{3}$. To combine the 'x' terms, we need to add their coefficients. This involves finding a common denominator for the fractions $\frac{1}{6}$ and $\frac{2}{3}$. The least common denominator (LCD) for 6 and 3 is 6. So, we rewrite $-\frac{2}{3}$ as $-\frac{4}{6}$. Now we can add the coefficients: $\frac{1}{6}x - \frac{4}{6}x = -\frac{3}{6}x$. This simplifies to $-\frac{1}{2}x$. Next, we combine the constant terms: $\frac{1}{3} - \frac{2}{3} = -\frac{1}{3}$. Now, we have simplified the expression inside the parentheses to $-\frac{1}{2}x - \frac{1}{3}$. This process of combining like terms is a cornerstone of algebraic simplification. It allows us to reduce the number of terms in an expression, making it easier to work with. By carefully identifying and combining like terms, we can significantly simplify complex expressions. This step lays the foundation for the next stage of simplification, where we will apply the distributive property. Understanding how to combine like terms is crucial for success in algebra and beyond.

Step 2: Applying the Distributive Property

Now that we've simplified the expression inside the parentheses to $-\frac{1}{2}x - \frac{1}{3}$, the next step is to apply the distributive property. The distributive property states that $a(b + c) = ab + ac$. In our case, we have $12(-\frac{1}{2}x - \frac{1}{3})$. This means we need to multiply 12 by each term inside the parentheses. First, we multiply 12 by $-\frac{1}{2}x$: $12 * (-\frac{1}{2}x) = -6x$. Next, we multiply 12 by $-\frac{1}{3}$: $12 * (-\frac{1}{3}) = -4$. So, after applying the distributive property, our expression becomes $-6x - 4$. The distributive property is a fundamental concept in algebra, allowing us to expand expressions and eliminate parentheses. It's essential for simplifying expressions and solving equations. By carefully applying the distributive property, we can transform a complex expression into a simpler one. This step is crucial for isolating variables and solving for unknowns in algebraic equations. Understanding and mastering the distributive property is a key skill for any algebra student. It's used extensively in various algebraic manipulations and is a building block for more advanced mathematical concepts. By practicing and applying the distributive property, you can gain confidence in your algebraic abilities and simplify a wide range of expressions.

The Simplified Expression: $-6x - 4$

After performing the steps of combining like terms and applying the distributive property, we arrive at the simplified expression: $-6x - 4$. This expression is equivalent to the original expression $12(\frac{1}{6}x + \frac{1}{3} - \frac{2}{3}x - \frac{2}{3})$, but it is in a much simpler form. The simplified expression is easier to understand and work with, making it more convenient for solving equations or further algebraic manipulations. This final form clearly shows the relationship between the variable 'x' and the constant term. The coefficient -6 indicates the rate of change of the expression with respect to 'x', and the constant term -4 represents the value of the expression when x is zero. Understanding the components of a simplified expression is crucial for interpreting its meaning and applying it in various contexts. The process of simplifying expressions is not just about finding a shorter form; it's about gaining a deeper understanding of the underlying mathematical relationships. By simplifying expressions, we can reveal the core structure of an equation or formula, making it easier to analyze and use. This skill is essential for success in algebra and higher-level mathematics. The ability to simplify expressions efficiently and accurately is a valuable asset in problem-solving and critical thinking.

Conclusion: Mastering Algebraic Simplification

In conclusion, we have successfully simplified the expression $12(\frac{1}{6}x + \frac{1}{3} - \frac{2}{3}x - \frac{2}{3})$ to its equivalent form, $-6x - 4$. This process involved two key steps: combining like terms within the parentheses and applying the distributive property. These techniques are fundamental to algebraic simplification and are essential skills for anyone studying mathematics. Mastering these skills not only allows us to simplify complex expressions but also provides a deeper understanding of algebraic principles. The ability to simplify expressions is crucial for solving equations, graphing functions, and tackling more advanced mathematical concepts. By breaking down complex expressions into simpler forms, we can make them more manageable and easier to analyze. This process enhances our problem-solving abilities and allows us to approach mathematical challenges with confidence. Algebraic simplification is a foundational skill that builds a strong base for further mathematical studies. It's a skill that is applicable in various fields, from science and engineering to finance and economics. By practicing and mastering algebraic simplification, you are equipping yourself with a valuable tool that will serve you well in your academic and professional pursuits. Remember, the key to success in algebra is practice and a thorough understanding of the underlying principles. Keep practicing, and you'll be well on your way to mastering algebraic simplification.