Simplify 7(x-2)-4x+9 A Step-by-Step Guide
Simplifying algebraic expressions is a fundamental skill in mathematics. In this comprehensive guide, we will delve into the process of simplifying the expression 7(x-2)-4x+9, breaking down each step to ensure clarity and understanding. This skill is crucial for solving equations, understanding functions, and tackling more advanced mathematical concepts. We will not only provide the correct answer but also explain the reasoning behind each step, offering a robust understanding of the underlying principles. By mastering this technique, you'll be better equipped to handle various algebraic manipulations, making your journey through mathematics smoother and more rewarding. Let's begin by understanding the core concepts involved in simplifying expressions, such as the distributive property and combining like terms, which are the cornerstones of this process.
Understanding the Basics of Algebraic Simplification
Before we jump into the specifics of simplifying 7(x-2)-4x+9, it's important to establish a solid understanding of the basic principles that govern algebraic manipulations. Algebraic simplification is essentially the process of reducing an algebraic expression to its simplest form without changing its value. This often involves applying the order of operations, the distributive property, and combining like terms. The order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which operations should be performed. The distributive property allows us to multiply a single term by two or more terms inside a set of parentheses. Combining like terms, which are terms that have the same variable raised to the same power, is a key step in reducing the expression. These foundational concepts are not just applicable to this specific problem but are universally relevant in algebra and beyond. Mastering these basics will significantly enhance your ability to simplify a wide range of expressions, making complex problems more manageable and less intimidating. As we proceed, we will see how these principles come into play in simplifying the given expression, illustrating their practical application and reinforcing their importance.
Step-by-Step Simplification of 7(x-2)-4x+9
To simplify the expression 7(x-2)-4x+9, we'll follow a systematic, step-by-step approach, ensuring that each operation is performed correctly and the reasoning behind it is clear. This methodical approach is essential for accuracy and for building a strong understanding of the process. Our first step involves applying the distributive property to the term 7(x-2). This means multiplying 7 by both x and -2 inside the parentheses. This step is crucial because it eliminates the parentheses, allowing us to combine like terms later on. Once we've applied the distributive property, we'll have a new expression with individual terms that are easier to manage. The next step is to identify and combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, the terms involving 'x' are like terms, and the constant terms are like terms. Combining them simplifies the expression further. By following these steps carefully, we'll arrive at the simplified form of the original expression. This process not only gives us the correct answer but also provides a clear pathway for simplifying similar expressions in the future. Let's now proceed with the actual simplification, breaking down each step in detail.
1. Applying the Distributive Property
The first crucial step in simplifying 7(x-2)-4x+9 is to apply the distributive property. This property states that a(b + c) = ab + ac. In our case, we need to distribute the 7 across the terms inside the parentheses (x and -2). This means we multiply 7 by x and 7 by -2. This step is fundamental because it removes the parentheses, allowing us to combine like terms in the subsequent steps. The distributive property is a cornerstone of algebraic manipulation, and understanding its application is vital for simplifying expressions effectively. It essentially transforms a more complex structure (an expression within parentheses multiplied by a term) into a simpler form (individual terms that can be more easily combined or manipulated). When we multiply 7 by x, we get 7x. When we multiply 7 by -2, we get -14. So, after applying the distributive property, our expression becomes 7x - 14 - 4x + 9. This transformed expression is now ready for the next step in our simplification process, which involves identifying and combining like terms. Let's proceed to that step to further reduce the expression.
2. Combining Like Terms
After applying the distributive property, our expression is now 7x - 14 - 4x + 9. The next step in simplifying this expression is to 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' (7x and -4x) and two constant terms (-14 and +9). Combining like terms involves adding or subtracting the coefficients (the numerical part of the term) of the terms with the same variable and adding or subtracting the constant terms. This process is essential for reducing the expression to its simplest form. It consolidates the terms, making the expression more concise and easier to understand. When we combine 7x and -4x, we subtract the coefficients (7 - 4) to get 3x. When we combine -14 and +9, we add them together (-14 + 9) to get -5. Therefore, after combining like terms, our expression simplifies to 3x - 5. This simplified form is the final result of our algebraic manipulation. Let's summarize the steps we've taken to arrive at this solution.
Final Simplified Expression and Answer
After meticulously applying the distributive property and combining like terms, we have successfully simplified the expression 7(x-2)-4x+9. The final simplified expression is 3x - 5. This process involved two key steps: first, distributing the 7 across the terms inside the parentheses, resulting in 7x - 14 - 4x + 9; and second, combining the like terms (7x and -4x, -14 and +9) to arrive at the simplified form. The simplified expression, 3x - 5, is the most concise representation of the original expression, maintaining its mathematical equivalence while being easier to understand and work with. This simplification not only provides the answer but also demonstrates a fundamental algebraic technique that is applicable in various mathematical contexts. The ability to simplify expressions is a crucial skill for solving equations, analyzing functions, and tackling more complex algebraic problems. Therefore, understanding the process and the reasoning behind each step is essential for mastering algebra and related mathematical disciplines. In the next section, we'll discuss some common mistakes to avoid when simplifying expressions.
Common Mistakes to Avoid When Simplifying Expressions
Simplifying expressions, while a fundamental skill, can be prone to errors if not approached with careful attention to detail. Recognizing and avoiding these common mistakes is crucial for ensuring accuracy and building confidence in your algebraic abilities. One frequent error is the incorrect application of the distributive property. For instance, forgetting to distribute the term to all elements inside the parentheses or miscalculating the signs during multiplication are common pitfalls. Another prevalent mistake is incorrectly combining like terms. This can occur when terms with different variables or exponents are mistakenly combined, or when the coefficients are added or subtracted incorrectly. It's also important to adhere strictly to the order of operations (PEMDAS). Deviating from this order can lead to incorrect results, especially when dealing with more complex expressions involving multiple operations. Additionally, sign errors are a common source of mistakes. A misplaced negative sign can drastically alter the outcome of the simplification. To mitigate these risks, it's helpful to double-check each step, pay close attention to signs and coefficients, and ensure that like terms are correctly identified before combining them. Practicing a variety of simplification problems can also help reinforce the correct techniques and build the habit of careful and accurate execution. By being aware of these common errors and actively working to avoid them, you can significantly improve your ability to simplify algebraic expressions effectively.
Conclusion: Mastering Algebraic Simplification
In conclusion, simplifying algebraic expressions is a foundational skill in mathematics that underpins many more advanced concepts. Throughout this guide, we have meticulously broken down the process of simplifying the expression 7(x-2)-4x+9, illustrating each step with clarity and precision. We began by understanding the basic principles of algebraic simplification, including the order of operations, the distributive property, and the concept of like terms. We then applied these principles step-by-step to the given expression, first distributing the 7 across the terms inside the parentheses and then combining like terms to arrive at the simplified form, 3x - 5. This methodical approach not only provided us with the correct answer but also demonstrated a clear and replicable process for simplifying similar expressions. Furthermore, we highlighted common mistakes to avoid during simplification, such as misapplying the distributive property, incorrectly combining like terms, and overlooking sign errors. By being aware of these pitfalls and practicing careful execution, you can significantly enhance your accuracy and confidence in algebraic manipulations. Mastering algebraic simplification is not just about finding the right answer; it's about developing a deeper understanding of the underlying mathematical principles and building a solid foundation for future learning in mathematics and related fields. With consistent practice and attention to detail, you can confidently tackle a wide range of algebraic simplification problems and unlock the doors to more advanced mathematical concepts.
