Simplifying Algebraic Expressions A Step-by-Step Guide To 7x - 3(-5y + 4x) + 3y

Understanding the Importance of Simplifying Expressions

In the realm of mathematics, simplifying expressions is a fundamental skill that lays the groundwork for more advanced concepts. At its core, simplification involves taking a complex expression and rewriting it in a more manageable and concise form. This not only makes the expression easier to understand but also facilitates further calculations and problem-solving. When we talk about the importance of simplifying expressions, it's not just about aesthetics; it's about practicality. Imagine trying to solve an equation with a needlessly complicated expression – the chances of making errors increase significantly. By simplifying, we reduce the complexity and the likelihood of mistakes, thereby enhancing the accuracy of our results. Moreover, simplified expressions often reveal the underlying structure and relationships within a mathematical problem, providing valuable insights that might otherwise remain hidden. In various fields, from physics to engineering to computer science, the ability to simplify expressions is crucial. Think about designing a bridge, writing a computer program, or modeling a physical system – all these tasks require the manipulation and simplification of mathematical expressions to arrive at effective solutions. The ability to simplify expressions efficiently and accurately is a hallmark of mathematical proficiency. It’s a skill that's honed through practice and a solid grasp of algebraic principles. As we delve deeper into mathematics, the expressions we encounter will undoubtedly become more intricate, making simplification an indispensable tool in our arsenal. Therefore, mastering this skill early on is not just beneficial; it's essential for long-term success in mathematics and related disciplines.

Breaking Down the Given Expression: 7x - 3(-5y + 4x) + 3y

Let's take a closer look at the expression we're going to simplify: 7x - 3(-5y + 4x) + 3y. This expression involves variables (x and y), coefficients (the numbers multiplying the variables), and constants. To effectively simplify this, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). The first step in our journey to simplify is to address the parentheses. We have -3(-5y + 4x), which means we need to distribute the -3 to both terms inside the parentheses. This is a crucial step because it removes the parentheses and allows us to combine like terms later. Distributing -3 to -5y gives us +15y, and distributing -3 to +4x gives us -12x. So, the expression now looks like this: 7x + 15y - 12x + 3y. Notice how the multiplication has transformed the terms inside the parentheses and set us up for the next stage of simplification. The distributive property is a cornerstone of algebraic manipulation, and its correct application is key to arriving at the right answer. Understanding how the negative sign affects the terms inside the parentheses is also vital, as it's a common source of errors. By carefully applying the distributive property, we've successfully eliminated the parentheses and laid the groundwork for combining like terms. This step-by-step approach ensures that we maintain accuracy and avoid rushing through the simplification process. With the parentheses gone, we're now in a position to group and combine the like terms, bringing us closer to the fully simplified expression. The expression 7x - 3(-5y + 4x) + 3y may have initially seemed complex, but by breaking it down and addressing each part systematically, we're making progress toward a more concise and understandable form.

Applying the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions involving parentheses. It states that a(b + c) = ab + ac. In simpler terms, it means that we can multiply a term outside the parentheses by each term inside the parentheses. In our expression, 7x - 3(-5y + 4x) + 3y, the key part where we need to apply the distributive property is the term -3(-5y + 4x). Here, -3 is the term outside the parentheses, and -5y and 4x are the terms inside. To apply the distributive property, we multiply -3 by each term inside the parentheses separately. First, we multiply -3 by -5y. A negative times a negative results in a positive, so -3 * -5y = 15y. Next, we multiply -3 by 4x. A negative times a positive results in a negative, so -3 * 4x = -12x. Now, we can rewrite the original expression by replacing -3(-5y + 4x) with 15y - 12x. This gives us a new expression: 7x + 15y - 12x + 3y. By carefully applying the distributive property, we've expanded the expression and removed the parentheses, which is a crucial step in simplifying it. It’s important to pay close attention to the signs when distributing, as a simple sign error can change the entire result. The distributive property is not just a mathematical rule; it’s a tool that helps us break down complex expressions into smaller, more manageable parts. By mastering this property, we gain a deeper understanding of how algebraic expressions work and how to manipulate them effectively. The ability to accurately apply the distributive property is essential for simplifying a wide range of algebraic expressions, making it a cornerstone of algebraic proficiency. As we continue to simplify our expression, the removal of parentheses through the distributive property has paved the way for the next step: combining like terms.

Combining Like Terms for Simplification

After applying the distributive property, our expression now reads: 7x + 15y - 12x + 3y. The next crucial 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 -12x) and two terms with the variable 'y' (15y and 3y). To combine like terms, we simply add or subtract their coefficients (the numbers in front of the variables). Let's start with the 'x' terms: 7x - 12x. To combine these, we subtract 12 from 7, which gives us -5. So, 7x - 12x = -5x. Now, let's move on to the 'y' terms: 15y + 3y. To combine these, we add 15 and 3, which gives us 18. So, 15y + 3y = 18y. Now that we've combined the 'x' terms and the 'y' terms, we can rewrite the expression as -5x + 18y. This is the simplified form of the original expression. By combining like terms, we've reduced the number of terms in the expression, making it more concise and easier to understand. The process of combining like terms is a fundamental skill in algebra, and it's essential for simplifying a wide variety of expressions. It’s a process that requires careful attention to the signs and coefficients of the terms. A solid understanding of combining like terms not only simplifies expressions but also lays the groundwork for solving equations and tackling more complex algebraic problems. The ability to accurately combine like terms is a hallmark of algebraic fluency and a crucial step in mastering mathematical simplification. Our journey from the initial expression to the final simplified form demonstrates the power of breaking down a problem into manageable steps and applying algebraic principles systematically.

Final Simplified Expression: -5x + 18y

After meticulously applying the distributive property and combining like terms, we've successfully simplified the original expression, 7x - 3(-5y + 4x) + 3y. The journey began with understanding the importance of simplification, then we carefully broke down the expression, addressed the parentheses using the distributive property, and finally combined like terms to arrive at our final answer. The simplified expression is -5x + 18y. This expression is more concise and easier to work with than the original. It clearly shows the relationship between the variables x and y, and it's in a standard algebraic form. The process of simplification is not just about finding the answer; it's about gaining a deeper understanding of the underlying mathematical structure. By simplifying, we've made the expression more transparent and accessible. This is particularly important when dealing with more complex problems where simplification is often a necessary step before further calculations can be made. The ability to simplify expressions efficiently and accurately is a valuable skill in mathematics and many other fields. It allows us to solve problems more effectively and to communicate mathematical ideas more clearly. The final simplified expression, -5x + 18y, represents the culmination of our efforts and a testament to the power of algebraic manipulation. It's a clear and concise representation of the original expression, ready for further use in any mathematical context. Through this process, we've reinforced our understanding of key algebraic principles and honed our simplification skills. Mastering simplification is an ongoing journey, and each expression we simplify brings us closer to a deeper understanding of mathematics.