Simplifying (35-7x) + 9x(2x-3) + 8 A Step-by-Step Guide
In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. This article delves into the process of simplifying the expression (35-7x) + 9x(2x-3) + 8, providing a step-by-step guide suitable for learners of all levels. We will break down each step, ensuring clarity and understanding, and also discuss the underlying principles that govern algebraic simplification. This comprehensive guide aims to empower you with the tools and knowledge necessary to tackle similar algebraic challenges with confidence. Mastering this skill is crucial for success in higher-level mathematics, making it an invaluable asset for students and enthusiasts alike.
Understanding the Basics of Algebraic Expressions
Before we dive into the specifics of simplifying (35-7x) + 9x(2x-3) + 8, let's establish a firm understanding of the basic concepts. Algebraic expressions are combinations of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Variables are symbols, typically letters like x or y, that represent unknown values. Constants, on the other hand, are fixed numerical values, such as 35, 7, or 8 in our expression. The operations connect these elements, dictating how they interact with each other. To effectively simplify an algebraic expression, it's essential to grasp the order of operations (PEMDAS/BODMAS) and the distributive property, which we will explore in detail throughout this guide. Understanding these fundamentals lays the groundwork for confidently tackling more complex expressions and algebraic manipulations. Without a solid grasp of these principles, the process of simplification can become confusing and error-prone. Therefore, we'll revisit these concepts throughout our explanation, ensuring a comprehensive understanding.
The Order of Operations (PEMDAS/BODMAS)
The order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), is a set of rules that dictate the sequence in which mathematical operations should be performed. This order ensures that we arrive at the correct result when simplifying expressions. For example, in the expression (35-7x) + 9x(2x-3) + 8, we must first address the operations within the parentheses and any multiplications before we can perform additions or subtractions. Neglecting the order of operations can lead to incorrect simplifications and ultimately, wrong answers. The order of operations is not just a convention; it's a fundamental rule that ensures consistency and accuracy in mathematical calculations. It's like a grammar for mathematics, providing a structure that allows us to communicate and solve problems effectively. In our specific expression, PEMDAS guides us to first distribute the 9x term into the parentheses before combining like terms. This seemingly small detail is crucial for arriving at the correct simplified form. Mastering PEMDAS is, therefore, not just about memorizing an acronym; it's about understanding the logical flow of mathematical operations and applying it consistently.
The Distributive Property
The distributive property is a fundamental concept in algebra that allows us to multiply a single term by a group of terms inside parentheses. This property states that a( b + c) = ab + ac. In other words, the term outside the parentheses (a) is multiplied by each term inside the parentheses (b and c). This property is crucial for simplifying expressions like (35-7x) + 9x(2x-3) + 8 because it allows us to eliminate the parentheses and combine like terms. Without the distributive property, we would be unable to proceed with the simplification process. The distributive property is not just a mathematical trick; it's a powerful tool that stems from the basic principles of multiplication and addition. It allows us to break down complex expressions into smaller, more manageable parts. In the context of our example, the distributive property allows us to multiply 9x by both 2x and -3, which is a critical step in simplifying the overall expression. Understanding and applying the distributive property correctly is essential for accurate algebraic manipulation. It is a cornerstone of algebra and a skill that will be used repeatedly in more advanced mathematical concepts.
Step-by-Step Simplification of (35-7x) + 9x(2x-3) + 8
Now, let's embark on the journey of simplifying the expression (35-7x) + 9x(2x-3) + 8 step by step. We'll break down each operation, explaining the rationale behind it and ensuring that you understand the process thoroughly. Our goal is not just to arrive at the final answer but to equip you with the ability to simplify similar expressions independently. We will revisit the order of operations and the distributive property as we go, demonstrating their practical application in this specific example. By following these steps meticulously, you'll gain a deeper understanding of algebraic simplification and build the confidence to tackle more complex problems. This step-by-step approach is designed to make the process accessible and understandable, even for those who are new to algebra. So, let's roll up our sleeves and begin simplifying!
Step 1: Applying the Distributive Property
The first crucial step in simplifying (35-7x) + 9x(2x-3) + 8 is to apply the distributive property. As we discussed earlier, this property allows us to multiply the term outside the parentheses (9x) by each term inside the parentheses (2x and -3). So, we multiply 9x by 2x, which gives us 18x², and then multiply 9x by -3, which results in -27x. This transforms our expression into 35 - 7x + 18x² - 27x + 8. The distributive property is essential here because it eliminates the parentheses, allowing us to combine like terms in the subsequent steps. Without this step, the expression would remain in a less simplified form. It's important to remember that the distributive property is not just a mechanical process; it's a reflection of the fundamental properties of multiplication and addition. By understanding why it works, you can apply it more confidently and accurately. This step sets the stage for further simplification and brings us closer to the final simplified expression.
Step 2: Identifying and Combining Like Terms
After applying the distributive property, our expression stands as 35 - 7x + 18x² - 27x + 8. Now, the next step is to identify and combine like terms. Like terms are those that have the same variable raised to the same power. In our expression, we have constant terms (35 and 8), x terms (-7x and -27x), and an x² term (18x²). To combine like terms, we simply add or subtract their coefficients. Let's start with the constant terms: 35 + 8 = 43. Next, we combine the x terms: -7x - 27x = -34x. The x² term, 18x², has no other like terms, so it remains as is. Combining these results, we get the simplified expression 18x² - 34x + 43. This step is crucial because it reduces the number of terms in the expression, making it simpler and easier to work with. Combining like terms is a fundamental algebraic skill, and mastering it is essential for simplifying more complex expressions. It's like organizing a cluttered room – by grouping similar items together, we make the overall space more manageable and understandable.
Step 3: Arranging the Terms in Standard Form
Following the combination of like terms, our expression is now 18x² - 34x + 43. Although this is a simplified form, it's customary to arrange the terms in standard form. In standard form, the terms are arranged in descending order of their exponents. This means that the term with the highest exponent comes first, followed by the term with the next highest exponent, and so on, with the constant term coming last. In our case, the term with the highest exponent is 18x², followed by -34x, and then the constant term 43. Thus, the expression is already in standard form. Arranging expressions in standard form is not just a matter of convention; it makes it easier to compare and manipulate expressions. It provides a consistent and organized way of writing algebraic expressions, which is particularly important in more advanced mathematics. Thinking of it like alphabetizing a list – it provides a clear and consistent structure that makes things easier to find and compare. While our expression was already in standard form, it's an important step to consider in general when simplifying algebraic expressions. This step ensures clarity and consistency in mathematical communication.
Final Simplified Expression
After meticulously applying the distributive property, combining like terms, and arranging the terms in standard form, we arrive at the final simplified expression: 18x² - 34x + 43. This is the most concise and organized form of the original expression, (35-7x) + 9x(2x-3) + 8. The simplification process has transformed a seemingly complex expression into a more manageable quadratic expression. This final form is not only easier to understand but also facilitates further mathematical operations, such as solving for x or graphing the expression. The journey from the initial expression to this simplified form highlights the power of algebraic manipulation. Each step, from applying the distributive property to combining like terms, played a crucial role in reaching the final result. This process demonstrates the elegance and efficiency of mathematical principles in simplifying complex problems. The simplified expression, 18x² - 34x + 43, is a testament to the effectiveness of these algebraic techniques.
Common Mistakes to Avoid
Simplifying algebraic expressions can be tricky, and it's easy to make mistakes if you're not careful. One common mistake is forgetting the order of operations. For instance, someone might try to add terms before applying the distributive property, leading to an incorrect result. Another frequent error is incorrectly distributing the negative sign. When multiplying a term by a negative expression in parentheses, it's crucial to remember to distribute the negative sign to all terms inside the parentheses. For example, in our expression, failing to correctly distribute the 9x term would result in an incorrect simplification. Combining unlike terms is another pitfall. Only terms with the same variable and exponent can be combined. Mixing up these rules can lead to significant errors. Additionally, arithmetic errors in basic addition, subtraction, multiplication, or division can derail the entire simplification process. To avoid these mistakes, it's essential to practice diligently, double-check your work, and pay close attention to the details of each step. Understanding the underlying principles of algebra, such as the order of operations and the distributive property, is also crucial. By being mindful of these common pitfalls, you can improve your accuracy and confidence in simplifying algebraic expressions.
Practice Problems
To solidify your understanding of simplifying algebraic expressions, it's essential to practice. Here are a few practice problems similar to the one we've just worked through:
- Simplify: 2(x + 3) - 4x + 7
- Simplify: 5y - 3(2y - 1) + 4
- Simplify: (4a - 2) + 6a(a + 1) - 3
These problems will allow you to apply the steps and principles we've discussed in this guide. Remember to follow the order of operations, apply the distributive property where necessary, and combine like terms. Working through these practice problems will not only reinforce your understanding but also help you identify any areas where you may need further clarification. The key to mastering algebraic simplification is consistent practice and attention to detail. Don't be discouraged if you encounter challenges along the way; mistakes are a natural part of the learning process. By working through these problems and reviewing your solutions, you'll gradually develop your skills and confidence in simplifying algebraic expressions. So, grab a pen and paper, and let's put your newfound knowledge to the test!
Conclusion
In conclusion, simplifying algebraic expressions is a fundamental skill in mathematics. Through our detailed step-by-step guide, we've explored the process of simplifying the expression (35-7x) + 9x(2x-3) + 8, highlighting the importance of the order of operations, the distributive property, and combining like terms. We've also discussed common mistakes to avoid and provided practice problems to reinforce your understanding. Mastering these techniques is crucial for success in algebra and beyond. The ability to simplify expressions not only makes mathematical problems more manageable but also enhances your problem-solving skills in general. As you continue your mathematical journey, remember that practice and perseverance are key. With a solid understanding of the principles and consistent effort, you'll be well-equipped to tackle increasingly complex algebraic challenges. We hope this guide has provided you with the tools and knowledge you need to confidently simplify algebraic expressions and excel in your mathematical endeavors.
