Simplifying Algebraic Expressions A Step-by-Step Guide To 1.5(3x+6) + 0.5(3-x)
Introduction: Mastering Algebraic Simplification
Algebraic expressions can seem daunting at first glance, but with a systematic approach, they become manageable and even enjoyable to solve. In this comprehensive guide, we'll break down the process of simplifying the expression 1.5(3x+6) + 0.5(3-x), providing you with a step-by-step walkthrough and valuable insights along the way. Whether you're a student tackling homework or someone looking to brush up on your algebra skills, this article will equip you with the knowledge and confidence to tackle similar problems. So, let's dive in and unravel the mysteries of algebraic simplification! Our journey will begin with understanding the fundamental principles that govern algebraic manipulations, ensuring you have a solid foundation before we even start crunching numbers. We'll explore the distributive property, the concept of combining like terms, and the order of operations – all crucial elements in simplifying any algebraic expression. Remember, guys, algebra isn't about memorizing rules; it's about understanding the logic behind them. Once you grasp these core concepts, you'll find that simplifying expressions becomes less of a chore and more of an intriguing puzzle. This expression, 1.5(3x+6) + 0.5(3-x), serves as an excellent example to illustrate these principles in action. By the end of this article, you'll not only be able to simplify this particular expression but also apply the same techniques to a wide variety of algebraic problems. This understanding is fundamental for success in higher mathematics and various fields that rely on mathematical modeling, from engineering to economics. So, grab your pencils, open your notebooks, and let's embark on this algebraic adventure together! We'll transform this seemingly complex expression into a simplified, elegant form, showcasing the power and beauty of mathematics.
Step 1: Applying the Distributive Property
The distributive property is a cornerstone of algebraic simplification, and it's our first tool in tackling the expression 1.5(3x+6) + 0.5(3-x). This property allows us to multiply a term outside the parentheses by each term inside the parentheses. Think of it like sharing the love (or the multiplication) equally among the terms within the group. In our case, we have two instances where we need to apply the distributive property: 1.5 needs to be multiplied by both 3x and 6, and 0.5 needs to be multiplied by both 3 and -x. Let's break it down step by step to ensure clarity and accuracy. First, we'll distribute 1.5 across the terms inside the first set of parentheses (3x + 6). This means we'll multiply 1.5 by 3x and then 1.5 by 6. The calculation looks like this: 1. 5 * 3x = 4.5x 2. 5 * 6 = 9. So, the first part of our expression, 1.5(3x + 6), simplifies to 4.5x + 9. Now, let's move on to the second part of the expression, 0.5(3 - x). We'll distribute 0. 5 across the terms inside the parentheses (3 and -x). Remember, it's crucial to pay attention to the signs! We'll multiply 0.5 by 3 and then 0.5 by -x. The calculation is as follows: 1. 5 * 3 = 1.5 2. 5 * -x = -0.5x. Therefore, the second part of our expression, 0.5(3 - x), simplifies to 1.5 - 0.5x. Now, we've successfully applied the distributive property to both parts of the expression. Our expression, 1.5(3x+6) + 0.5(3-x), has been transformed into 4.5x + 9 + 1.5 - 0.5x. We've essentially cleared the parentheses and paved the way for the next crucial step: combining like terms. This step will bring us closer to the simplified form of the expression, making it easier to understand and work with. Remember, the distributive property is a fundamental skill in algebra, so mastering it is essential for success in simplifying more complex expressions. Keep practicing, and you'll become a pro in no time! So, guys, are you ready to move on to the next step and see how we can further simplify this expression? Let's do it!
Step 2: Combining Like Terms
After applying the distributive property, we've arrived at the expression 4.5x + 9 + 1.5 - 0.5x. The next crucial step in simplifying this expression is combining like terms. But what exactly are “like terms”? In algebra, like terms are those that have the same variable raised to the same power. They can be combined because they represent the same kind of quantity. Think of it like this: you can add apples to apples and oranges to oranges, but you can't directly add apples to oranges. Similarly, in our expression, the terms with 'x' (4.5x and -0.5x) are like terms, and the constant terms (9 and 1.5) are also like terms. We can combine the 'x' terms together and the constant terms together to simplify the expression. Let's start by focusing on the 'x' terms: 4.5x and -0.5x. To combine them, we simply add their coefficients (the numbers in front of the 'x'). So, we have 4.5 + (-0.5) = 4. This means that 4.5x - 0.5x simplifies to 4x. Now, let's move on to the constant terms: 9 and 1.5. These are straightforward to combine; we simply add them together: 9 + 1.5 = 10.5. Now, we've combined all the like terms in our expression. We have 4x from combining the 'x' terms and 10.5 from combining the constant terms. This means our expression 4.5x + 9 + 1.5 - 0.5x simplifies to 4x + 10.5. Wow, guys, look how much simpler the expression is now! We've taken a longer expression with multiple terms and condensed it into a concise and easy-to-understand form. This is the power of combining like terms. It allows us to represent the same mathematical relationship in a more efficient way. Combining like terms is a fundamental skill in algebra, and it's used extensively in solving equations, simplifying expressions, and working with polynomials. By mastering this skill, you'll be well-equipped to tackle more complex algebraic problems. So, always remember to identify and combine like terms whenever you're simplifying an expression. It's a key step in making algebra less daunting and more manageable. Now that we've successfully combined like terms, we've reached the final simplified form of our expression. Are you ready to see the grand finale and appreciate the beauty of our simplified expression? Let's move on to the final step and celebrate our algebraic victory!
Step 3: Presenting the Simplified Expression
After diligently applying the distributive property and combining like terms, we've arrived at the simplified form of our expression: 4x + 10.5. This is the culmination of our efforts, and it represents the most concise and understandable way to express the original algebraic relationship. It's like taking a tangled mess of string and neatly unraveling it into a smooth, single strand. The original expression, 1.5(3x+6) + 0.5(3-x), may have seemed complex at first glance, but through our step-by-step simplification process, we've transformed it into a much more manageable form. This final expression, 4x + 10.5, is not only simpler but also easier to work with in further calculations or problem-solving scenarios. For instance, if we were given a value for 'x', we could easily substitute it into this simplified expression to find the corresponding value of the entire expression. This is much easier than substituting the value into the original, more complex expression. Moreover, the simplified form often reveals the underlying structure and properties of the expression more clearly. In this case, we can see that the expression represents a linear relationship, where the value increases by 4 for every unit increase in 'x', and the expression has a constant term of 10.5. Presenting the simplified expression is not just about arriving at the final answer; it's about showcasing the elegance and efficiency of mathematics. It's about demonstrating how complex expressions can be reduced to their simplest forms through the application of fundamental principles. When presenting your simplified expression, it's always a good idea to double-check your work to ensure accuracy. A small error in any of the steps can lead to an incorrect final result. Also, make sure your answer is presented in a clear and organized manner, making it easy for others to understand your solution. Guys, we've successfully navigated the world of algebraic simplification and emerged victorious! We've taken a seemingly complex expression and transformed it into a beautifully simple form. This is a testament to your hard work and dedication to understanding the principles of algebra. So, let's take a moment to appreciate the journey and the knowledge we've gained along the way. The ability to simplify algebraic expressions is a valuable skill that will serve you well in various mathematical endeavors. Now, you're equipped to tackle similar problems with confidence and expertise. Remember, practice makes perfect, so keep honing your skills and exploring the fascinating world of algebra. And that's it! We've reached the end of our algebraic adventure. Congratulations on mastering the simplification of the expression 1.5(3x+6) + 0.5(3-x)!
Conclusion: The Power of Simplification
In conclusion, we've successfully simplified the algebraic expression 1.5(3x+6) + 0.5(3-x) to its concise form: 4x + 10.5. This journey has highlighted the power and importance of algebraic simplification in mathematics. By applying the distributive property and combining like terms, we've transformed a complex expression into a simpler, more manageable one. This process not only makes the expression easier to understand but also facilitates further calculations and problem-solving. Simplification is a fundamental skill in algebra and is crucial for success in higher-level mathematics and various fields that rely on mathematical modeling. It allows us to express mathematical relationships in their most efficient and elegant forms, making them easier to analyze and interpret. Throughout this guide, we've emphasized the importance of understanding the underlying principles behind each step. It's not enough to simply memorize the rules; you need to grasp the logic behind them to apply them effectively in different situations. The distributive property, for instance, is not just a mechanical rule; it's a reflection of how multiplication interacts with addition and subtraction. Similarly, combining like terms is based on the idea that we can only add or subtract quantities that represent the same kind of thing. Guys, mastering these fundamental concepts will empower you to tackle a wide range of algebraic problems with confidence. You'll be able to simplify expressions, solve equations, and work with polynomials more effectively. Remember, practice is key to honing your skills in algebra. The more you practice simplifying expressions, the more comfortable and confident you'll become. Don't be afraid to make mistakes; they're valuable learning opportunities. Analyze your errors, understand why they occurred, and learn from them. The world of algebra is vast and fascinating, and there's always something new to learn. Embrace the challenge, explore different techniques, and enjoy the process of discovery. We hope this comprehensive guide has provided you with a solid foundation in simplifying algebraic expressions. Remember the steps we've covered: applying the distributive property, combining like terms, and presenting the simplified expression in a clear and organized manner. These skills will serve you well in your mathematical journey. So, go forth and conquer the world of algebra! Simplify expressions with confidence, solve equations with ease, and unlock the power of mathematics. And always remember, guys, that math can be fun and rewarding when approached with the right mindset and a willingness to learn. Thank you for joining us on this algebraic adventure!
