Simplifying Algebraic Expressions Step By Step Guide

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Simplifying algebraic expressions is a fundamental skill in mathematics, essential for solving equations, understanding relationships, and building a strong foundation for more advanced concepts. This article will guide you through the step-by-step process of simplifying a complex algebraic expression, providing clear explanations and examples along the way. In this comprehensive guide, we will break down the intricate process of simplifying the algebraic expression: 2x³y - [ 5/6x³y - 2/3x³y - ( 4/7x³y - 1/2x³y - 3/14x³y ) + 5/21x³y ] - 25/21x³y. This expression, with its multiple layers of parentheses, fractions, and terms, can seem daunting at first glance. However, by systematically applying the order of operations and the principles of combining like terms, we can effectively simplify it to its most basic form. This article will walk you through each step, explaining the reasoning and techniques involved, ensuring you gain a solid understanding of the simplification process. Algebraic expressions are the building blocks of algebra, and the ability to simplify them is crucial for success in higher-level mathematics. This guide aims to demystify the process, making it accessible to learners of all levels. We'll start by understanding the basic rules of simplifying expressions, then move on to the specific steps involved in tackling the given problem. By the end of this article, you'll not only be able to simplify this particular expression but also have the confidence and skills to tackle other complex algebraic expressions. The core of simplifying any algebraic expression lies in understanding the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which operations must be performed to arrive at the correct answer. For instance, operations within parentheses are always addressed first, followed by exponents, then multiplication and division (from left to right), and finally, addition and subtraction (also from left to right). Additionally, the concept of "like terms" is crucial. Like terms are those that have the same variables raised to the same powers. For example, 3x² and -5x² are like terms because they both have the variable x raised to the power of 2. Only like terms can be combined by adding or subtracting their coefficients (the numbers in front of the variables). With these fundamental principles in mind, let's dive into simplifying the given expression. We will meticulously work through each layer, breaking down the problem into manageable steps and explaining the logic behind each manipulation. This approach will not only help you understand the solution but also equip you with the skills to independently simplify similar expressions in the future. Simplifying algebraic expressions is not just about following a set of rules; it's about developing a mathematical intuition and a problem-solving mindset. This article is designed to foster that intuition, guiding you towards a deeper understanding of algebra and its applications. Let’s embark on this journey of simplification and unlock the beauty and power of algebraic manipulation.

Step-by-Step Simplification

To simplify this algebraic expression, we'll follow a step-by-step process, prioritizing operations within parentheses and working our way outwards. The expression we need to simplify is: 2x³y - [ 5/6x³y - 2/3x³y - ( 4/7x³y - 1/2x³y - 3/14x³y ) + 5/21x³y ] - 25/21x³y. The initial step involves focusing on the innermost parentheses. This is in line with the order of operations (PEMDAS/BODMAS), which prioritizes simplifying expressions within parentheses first. By addressing the innermost layer, we effectively reduce the complexity of the overall expression, making it easier to manage in subsequent steps. This approach is akin to peeling an onion, layer by layer, until we reach the core. Within the innermost parentheses, we have the expression (4/7x³y - 1/2x³y - 3/14x³y). Here, all terms are like terms because they share the same variables (x and y) raised to the same powers (3 and 1, respectively). This allows us to combine these terms by adding or subtracting their coefficients. To do this, we need to find a common denominator for the fractions 4/7, 1/2, and 3/14. The least common multiple (LCM) of 7, 2, and 14 is 14. So, we'll convert each fraction to have a denominator of 14. This conversion is a crucial step in ensuring accurate calculations, as it allows us to directly add and subtract the numerators. Once we have a common denominator, we can combine the fractions, simplifying the expression within the parentheses. This simplification not only reduces the number of terms but also makes the overall expression less cumbersome. The next step involves substituting the simplified expression back into the original equation and continuing the simplification process. This iterative approach, where we simplify one part of the expression at a time, is a key strategy in tackling complex algebraic problems. It allows us to break down a large problem into smaller, more manageable parts, reducing the likelihood of errors and making the solution process more transparent. Moreover, it reinforces the understanding of algebraic manipulation rules and techniques. As we progress through the steps, we'll continue to emphasize the importance of accurate arithmetic and careful attention to detail. Each step builds upon the previous one, so any error early on can propagate through the rest of the solution. Therefore, it's essential to double-check our work at each stage, ensuring that we're maintaining the integrity of the expression and moving towards the correct simplification. By following this systematic approach, we'll not only arrive at the final answer but also gain a deeper appreciation for the elegance and precision of algebra. Let's delve into the details of each step, starting with the simplification of the innermost parentheses. This journey through the expression will highlight the power of methodical problem-solving and the beauty of mathematical simplification.

Simplify Innermost Parentheses: (4/7x³y - 1/2x³y - 3/14x³y)

To simplify the innermost parentheses, we need to combine the like terms. The expression inside the innermost parentheses is (4/7x³y - 1/2x³y - 3/14x³y). As mentioned earlier, the key to combining these terms is to find a common denominator for the fractions. The fractions involved are 4/7, 1/2, and 3/14. To find the least common denominator (LCD), we need to determine the least common multiple (LCM) of the denominators 7, 2, and 14. The LCM of 7, 2, and 14 is 14. This means we will convert each fraction to an equivalent fraction with a denominator of 14. Converting fractions to a common denominator is a fundamental skill in arithmetic and algebra. It allows us to perform addition and subtraction operations on fractions accurately. The process involves multiplying both the numerator and the denominator of each fraction by a suitable number that will result in the desired common denominator. This ensures that we are not changing the value of the fraction, only its representation. For the fraction 4/7, we multiply both the numerator and the denominator by 2, resulting in 8/14. For the fraction 1/2, we multiply both the numerator and the denominator by 7, resulting in 7/14. The fraction 3/14 already has the desired denominator, so it remains unchanged. Now that we have a common denominator, we can rewrite the expression as (8/14x³y - 7/14x³y - 3/14x³y). The next step is to combine the numerators while keeping the common denominator. This is done by adding or subtracting the numerators as indicated by the signs in the expression. In this case, we have 8 - 7 - 3. Performing these operations, we get 8 - 7 = 1, and then 1 - 3 = -2. So, the combined numerator is -2. Therefore, the simplified fraction is -2/14. This fraction can be further simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2. Dividing -2 by 2 gives -1, and dividing 14 by 2 gives 7. So, the simplified fraction is -1/7. Substituting this back into the expression with the variables, we get -1/7x³y. This is the simplified form of the expression within the innermost parentheses. This process of simplifying fractions and combining like terms is a cornerstone of algebraic manipulation. It requires a solid understanding of fraction arithmetic and the ability to apply the rules of addition and subtraction to algebraic expressions. By mastering these skills, you'll be well-equipped to tackle more complex algebraic problems. Now that we have simplified the innermost parentheses, we can substitute this result back into the original expression and continue the simplification process. This step-by-step approach is crucial for maintaining accuracy and clarity in algebraic manipulations. Each simplification makes the expression more manageable, leading us closer to the final answer. Let's proceed with the next step and see how this simplification affects the overall expression.

Substitute and Simplify the Brackets

After simplifying the innermost parentheses, we substitute the result back into the original expression. The original expression was: 2x³y - [ 5/6x³y - 2/3x³y - ( 4/7x³y - 1/2x³y - 3/14x³y ) + 5/21x³y ] - 25/21x³y. We found that (4/7x³y - 1/2x³y - 3/14x³y) simplifies to -1/7x³y. Substituting this back into the original expression, we get: 2x³y - [ 5/6x³y - 2/3x³y - (-1/7x³y) + 5/21x³y ] - 25/21x³y. Notice the double negative in front of the term -1/7x³y. When we subtract a negative number, it's the same as adding its positive counterpart. So, - (-1/7x³y) becomes + 1/7x³y. This is a crucial step in maintaining the accuracy of the expression. The expression now becomes: 2x³y - [ 5/6x³y - 2/3x³y + 1/7x³y + 5/21x³y ] - 25/21x³y. Next, we need to simplify the expression within the square brackets. To do this, we need to combine the like terms inside the brackets. The terms are 5/6x³y, -2/3x³y, 1/7x³y, and 5/21x³y. To combine these terms, we need to find a common denominator for the fractions 5/6, -2/3, 1/7, and 5/21. The least common multiple (LCM) of 6, 3, 7, and 21 is 42. So, we will convert each fraction to an equivalent fraction with a denominator of 42. Converting each fraction, we get: * 5/6 = (5 * 7) / (6 * 7) = 35/42 * -2/3 = (-2 * 14) / (3 * 14) = -28/42 * 1/7 = (1 * 6) / (7 * 6) = 6/42 * 5/21 = (5 * 2) / (21 * 2) = 10/42 Now we can rewrite the expression within the brackets as: [ 35/42x³y - 28/42x³y + 6/42x³y + 10/42x³y ]. Combining the numerators, we get: 35 - 28 + 6 + 10 = 23. So, the simplified fraction is 23/42. Therefore, the expression within the brackets simplifies to 23/42x³y. Substituting this back into the main expression, we get: 2x³y - [ 23/42x³y ] - 25/21x³y. Now we can remove the square brackets, which gives us: 2x³y - 23/42x³y - 25/21x³y. This step of substituting and simplifying the brackets is crucial because it reduces the complexity of the expression and brings us closer to the final simplified form. It demonstrates the importance of working systematically from the innermost parentheses outwards, combining like terms, and paying close attention to signs. Now that we have a simpler expression, we can proceed to the next step, which involves combining the remaining like terms. This will lead us to the final simplified form of the algebraic expression. Let's continue the journey and see how the final steps unfold.

Combine Remaining Like Terms

To combine the remaining like terms, we now have the expression: 2x³y - 23/42x³y - 25/21x³y. All the terms in this expression are like terms because they have the same variables (x and y) raised to the same powers (3 and 1, respectively). This means we can combine them by adding or subtracting their coefficients. First, we need to find a common denominator for the fractions. The coefficients are 2 (which can be written as 2/1), -23/42, and -25/21. The least common multiple (LCM) of 1, 42, and 21 is 42. So, we will convert each fraction to an equivalent fraction with a denominator of 42. Converting each fraction, we get: * 2/1 = (2 * 42) / (1 * 42) = 84/42 * -23/42 remains as -23/42 * -25/21 = (-25 * 2) / (21 * 2) = -50/42 Now we can rewrite the expression as: 84/42x³y - 23/42x³y - 50/42x³y. Next, we combine the numerators while keeping the common denominator: (84 - 23 - 50) / 42. Performing the operations, we get: * 84 - 23 = 61 * 61 - 50 = 11 So, the combined numerator is 11. Therefore, the simplified fraction is 11/42. Substituting this back into the expression with the variables, we get 11/42x³y. This is the final simplified form of the original algebraic expression. This process of combining like terms is a fundamental skill in algebra. It involves identifying terms with the same variables and exponents, finding a common denominator for their coefficients, and then adding or subtracting the coefficients. This simplification process is crucial for solving equations, simplifying complex expressions, and making mathematical problems more manageable. By carefully following each step, we have successfully simplified the given algebraic expression. We started with a complex expression with multiple layers of parentheses and fractions, and through a systematic approach, we have reduced it to its simplest form. This demonstrates the power of algebraic manipulation and the importance of understanding the underlying principles. In conclusion, the simplified form of the expression 2x³y - [ 5/6x³y - 2/3x³y - ( 4/7x³y - 1/2x³y - 3/14x³y ) + 5/21x³y ] - 25/21x³y is 11/42x³y. This result showcases the elegance and efficiency of algebraic simplification, transforming a seemingly complex problem into a concise and understandable form.

Final Simplified Expression

Therefore, after carefully following the steps of simplifying algebraic expressions, the final simplified expression is:

11/42 x³y

This article has demonstrated the step-by-step process of simplifying a complex algebraic expression. By understanding the order of operations, combining like terms, and working systematically, we can effectively simplify even the most challenging expressions. This skill is essential for success in algebra and beyond.

  • Simplifying algebraic expressions
  • Algebraic expressions
  • Simplification
  • Mathematics
  • Step-by-step guide
  • Combining like terms
  • Order of operations
  • Fractions
  • Common denominator
  • Coefficients
  • PEMDAS
  • Solving equations
  • Algebra
  • Mathematical expressions
  • Simplifying fractions