Simplifying Expressions A Step-by-Step Guide To 2(4x-3)-6

In the realm of mathematics, simplifying expressions is a fundamental skill. It's the art of taking a complex-looking mathematical statement and reducing it to its most basic, understandable form. This ability is crucial for solving equations, understanding mathematical concepts, and even for real-world applications like budgeting or calculating measurements. In this comprehensive guide, we'll dissect the expression $2(4x-3)-6$, providing a detailed, step-by-step approach to simplification. Our goal is not just to arrive at the correct answer, but to equip you with the understanding and confidence to tackle similar problems on your own. Whether you're a student grappling with algebra or simply seeking to refresh your math skills, this article will serve as a valuable resource. Let's embark on this mathematical journey together, unraveling the complexities and revealing the elegant simplicity that lies within.

Understanding the Expression

Before we dive into the simplification process, it's crucial to first understand the expression we're dealing with: $2(4x-3)-6$. This expression involves several mathematical operations: multiplication, subtraction, and the distributive property. Let's break down each component to gain a clearer picture.

• The Variable: The expression contains the variable 'x'. In algebra, a variable represents an unknown value. Our goal in many algebraic problems is to find the value of this variable, or in this case, to simplify the expression containing it.
• Parentheses: The parentheses around $(4x-3)$ indicate that the terms inside should be treated as a single unit. This is where the distributive property comes into play, which we'll discuss shortly.
• Multiplication: The number 2 outside the parentheses is being multiplied by the entire expression inside the parentheses. This is a key operation in simplifying the expression.
• Subtraction: Finally, we have the subtraction of 6 from the result of the multiplication. This is the last operation we'll perform in our simplification process.

By understanding each of these components, we can develop a strategic approach to simplifying the expression. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), will guide us through the process. With this foundational understanding, we're ready to begin the simplification journey, applying the distributive property and combining like terms to reach our final answer.

Applying the Distributive Property

The distributive property is the cornerstone of simplifying expressions like $2(4x-3)-6$. This property allows us to multiply a single term by each term inside a set of parentheses. In essence, it breaks down a complex multiplication into simpler ones. Let's illustrate how it works in our specific expression.

The distributive property states that for any numbers a, b, and c:

$a(b + c) = ab + ac$

In our case, 'a' is 2, 'b' is 4x, and 'c' is -3. Applying the distributive property, we multiply 2 by both terms inside the parentheses:

$2(4x - 3) = (2 * 4x) + (2 * -3)$

Now, let's perform the multiplications:

• $2 * 4x = 8x$
• $2 * -3 = -6$

So, after applying the distributive property, the expression $2(4x-3)$ becomes $8x - 6$. It's crucial to pay attention to the signs when multiplying. A positive number multiplied by a negative number results in a negative number, which is why we have -6.

This step is a significant milestone in our simplification journey. We've successfully removed the parentheses and transformed the expression into a more manageable form. However, we're not quite at the final answer yet. We still have the -6 outside the parentheses to consider. In the next step, we'll combine like terms to complete the simplification process, building upon the foundation we've established with the distributive property. Understanding and correctly applying the distributive property is essential for success in algebra and beyond.

Combining Like Terms

After applying the distributive property, our expression stands as $8x - 6 - 6$. The next crucial step in simplifying is to combine like terms. This involves identifying terms that share the same variable and exponent (in this case, only the constant terms) and then adding or subtracting them as indicated.

In our expression, $8x - 6 - 6$, we have one term with a variable ($8x$) and two constant terms (-6 and -6). Like terms are terms that have the same variable raised to the same power. Constant terms, which are just numbers without any variables, are also considered like terms.

To combine the like terms, we focus on the constant terms, -6 and -6. We perform the subtraction:

$-6 - 6 = -12$

Now, we replace the original constant terms with their combined value. Our expression now looks like this:

$8x - 12$

We've successfully combined the like terms, resulting in a simplified expression. There are no more like terms to combine, and the expression is now in its most basic form. This step is crucial because it consolidates the expression, making it easier to understand and work with in further mathematical operations.

Combining like terms is a fundamental skill in algebra and is used extensively in solving equations and simplifying more complex expressions. By mastering this technique, you'll be well-equipped to tackle a wide range of mathematical problems. In the next section, we'll present the final simplified expression and discuss its implications, solidifying our understanding of the entire simplification process.

The Final Simplified Expression

After diligently applying the distributive property and combining like terms, we've arrived at the final simplified expression: $8x - 12$. This is the most concise and understandable form of the original expression, $2(4x-3)-6$. It represents the same mathematical relationship, but in a more streamlined and manageable way.

Let's recap the steps we took to reach this solution:

1. Applying the Distributive Property: We multiplied 2 by each term inside the parentheses: $2(4x-3) = 8x - 6$.
2. Combining Like Terms: We combined the constant terms: $-6 - 6 = -12$.

The result of these operations is $8x - 12$. This simplified expression is equivalent to the original, but it's easier to interpret and use in further calculations or problem-solving.

The simplified expression $8x - 12$ can be used in various contexts. For example, if we were given a value for 'x', we could easily substitute it into the simplified expression to find the numerical value of the expression. This is a common task in algebra and calculus.

Furthermore, the simplified expression reveals the underlying structure of the mathematical relationship. The term $8x$ indicates a linear relationship with a slope of 8, and the -12 represents the y-intercept if we were to graph this expression as a line. Understanding this structure can be invaluable in various mathematical and real-world applications.

In conclusion, simplifying expressions is a fundamental skill that allows us to represent mathematical relationships in their most basic form. By mastering techniques like the distributive property and combining like terms, we can tackle complex expressions and gain a deeper understanding of the underlying mathematical concepts. Our journey with the expression $2(4x-3)-6$ has demonstrated the power and elegance of simplification in mathematics.

Choosing the Correct Answer

Now that we've meticulously simplified the expression $2(4x-3)-6$ to $8x - 12$, let's revisit the multiple-choice options provided at the beginning and identify the correct answer. This step is crucial for solidifying our understanding and ensuring we can apply our simplification skills to answer questions accurately.

The multiple-choice options were:

A. $8x - 12$ B. $8x - 9$ C. $6x - 3$ D. $4x - 9$

By comparing our simplified expression, $8x - 12$, with the options, it's clear that option A, $8x - 12$, is the correct answer. The other options contain different coefficients for 'x' or incorrect constant terms, indicating errors in the simplification process.

Choosing the correct answer is not just about arriving at the right result; it's also about verifying our work and ensuring we haven't made any mistakes along the way. It's a final check that reinforces our understanding of the simplification process.

This exercise highlights the importance of accuracy and attention to detail in mathematics. A small error in applying the distributive property or combining like terms can lead to an incorrect answer. Therefore, it's always wise to double-check each step and compare the final simplified expression with the given options.

In this case, our step-by-step approach and careful execution have led us to the correct answer, demonstrating the effectiveness of a systematic approach to simplifying expressions. This skill is not only valuable for answering multiple-choice questions but also for tackling more complex mathematical problems in the future.

Conclusion: Mastering Expression Simplification

In this comprehensive guide, we've embarked on a journey to master the simplification of the expression $2(4x-3)-6$. We've broken down the process into manageable steps, from understanding the expression's components to applying the distributive property and combining like terms. Through this detailed exploration, we've not only arrived at the correct answer but also gained a deeper understanding of the underlying mathematical principles.

Simplifying expressions is a fundamental skill in algebra and mathematics as a whole. It's the cornerstone of solving equations, understanding mathematical relationships, and applying mathematical concepts to real-world scenarios. The ability to take a complex-looking expression and reduce it to its simplest form is a powerful tool in any mathematical toolkit.

We've covered two key techniques in this guide:

• The Distributive Property: This property allows us to multiply a single term by each term inside parentheses, effectively breaking down complex multiplications into simpler ones.
• Combining Like Terms: This involves identifying terms that share the same variable and exponent and then adding or subtracting them to consolidate the expression.

By mastering these techniques, you'll be well-equipped to tackle a wide range of simplification problems. Remember, practice is key to proficiency. The more you work with simplifying expressions, the more comfortable and confident you'll become.

Furthermore, understanding the order of operations (PEMDAS) is crucial for simplifying expressions correctly. This ensures that you perform operations in the right sequence, leading to accurate results.

In conclusion, simplifying expressions is not just about finding the right answer; it's about developing a deeper understanding of mathematical concepts and building problem-solving skills. By following the steps outlined in this guide and practicing regularly, you can master expression simplification and unlock new levels of mathematical proficiency. The journey to mathematical mastery is continuous, and simplifying expressions is a vital step along the way.