Simplifying Algebraic Expressions: A Step-by-Step Guide

by ADMIN 56 views

Hey everyone, let's dive into the world of simplifying algebraic expressions! These problems can seem a bit intimidating at first, but with a little practice and a clear understanding of the rules, you'll be cruising through them in no time. We're going to break down each expression step-by-step, making sure that you grasp the concepts. Get ready to flex those math muscles! This is a friendly guide, so don't worry if you're new to algebra. We'll cover everything you need to know, from the basics to more complex operations, ensuring you're well-equipped to tackle similar problems in the future. We'll be looking at how to simplify complex fractions, deal with variable manipulation, and manage order of operations to achieve the most simplified form of the given expressions. The ultimate goal is to transform a complicated algebraic expression into a cleaner, more manageable form. This isn’t just about getting the right answer, but understanding why the answer is what it is. This approach is all about empowering you with the knowledge to solve any algebraic problem you come across. So, let’s jump right in and make algebra fun! The goal is to arm you with the knowledge and skills needed to confidently solve these types of problems. By understanding the logic behind each step, you will build a solid foundation in algebra, enabling you to tackle more complex equations and concepts in the future. So, let's get started and make algebra less about memorization and more about understanding.

Expression 1: Simplify 1) 2y/mn : 4y/m²

Alright, let's start with our first expression: 2y/mn : 4y/m². Remember that the colon (:) indicates division. The key here is to understand how to handle division of fractions. The trick is to change the division operation into multiplication by flipping the second fraction (the divisor). This method of inverting and multiplying is a cornerstone of working with fractions, and knowing it cold will make your life much easier. With fractions, keeping track of each term is critical, especially when multiple variables and exponents are involved. Let's walk through this step-by-step to make sure everyone is on the same page, alright?

Firstly, rewrite the expression using the division rule, which says division is multiplication by the reciprocal: 2y/mn ÷ 4y/m² becomes 2y/mn * m²/4y. Now we have a multiplication problem, which is way more manageable. When multiplying fractions, you multiply the numerators together and the denominators together. Easy, right? This is like the bread and butter of working with fractions. Secondly, multiply the numerators: 2y * m² = 2ym². Then, multiply the denominators: mn * 4y = 4mny. The result is 2ym²/4mny. Now we're getting somewhere! Next, we simplify the expression. Look for common factors in the numerator and the denominator that can be canceled out. Let's look at the variables. The y in the numerator and the y in the denominator cancel out. We are left with 2m²/4mn. You can also simplify the numbers: 2/4 simplifies to 1/2. We now have m²/2mn. Lastly, cancel common factors. There's an m in the numerator (m² has two ms, so one m can be canceled) and an m in the denominator. This leaves us with m/2n. Therefore, the simplified form of 2y/mn : 4y/m² is m/2n. Pretty cool, huh? Remember, each step builds on the previous one. So, make sure you understand the process thoroughly before moving on. The entire process involves converting the division problem into a multiplication problem. Make sure you understand that concept, and you're well on your way to mastering these sorts of problems. Keep in mind, the more you practice these, the easier they will become. You've got this!

Summary of Steps for 1) 2y/mn : 4y/m²

  1. Rewrite Division: Convert division to multiplication by flipping the second fraction: 2y/mn ÷ 4y/m² -> 2y/mn * m²/4y
  2. Multiply Numerators: 2y * m² = 2ym²
  3. Multiply Denominators: mn * 4y = 4mny
  4. Result: 2ym²/4mny
  5. Simplify: Cancel y and simplify 2/4: m²/2mn
  6. Cancel: Cancel m: m/2n

Expression 2: Simplify 2) 3y + 3 / y - 2 × y - 2 / y² - y

Now, let's tackle something a bit more complicated: 3y + 3 / y - 2 × y - 2 / y² - y. Notice the order of operations here. We have to be very careful with how we interpret this expression. It's important to stick to the order of operations (PEMDAS/BODMAS) to ensure the correct solution. Without parentheses to clarify, we must work out the division and multiplication before addition and subtraction. So, we must do the division or multiplication first. It looks a bit like a jumbled mess, but breaking it down step by step will get us through it. Remember, when in doubt, simplify, and you'll get there. Taking it slow is key.

First, it helps to rewrite the expression more clearly, keeping the order of operations in mind: 3y + (3 / y) - 2 × (y - 2) / (y² - y). Now it's easier to see what needs to be done first. The division is between 3 and y, and the multiplication is between 2 and (y - 2) / (y² - y). Let's start by simplifying the multiplication part. Before we do that, let's factor the denominator y² - y. We can take out a common factor of y, which gives us y(y - 1). Now our expression looks like: 3y + (3 / y) - 2(y - 2) / y(y - 1). This is now a complex fraction and we need to simplify it. Next, we multiply the 2 with the numerator of the complex fraction, we have 2(y - 2) resulting (2y - 4) / y(y - 1). Hence, the expression becomes 3y + (3/y) - (2y - 4) / y(y - 1). Remember that we're trying to get a common denominator here so we can combine these terms, which is y(y - 1). Multiply the terms with what they need to have that common denominator, resulting 3y × y(y - 1) / y(y - 1) + 3(y - 1) / y(y - 1) - (2y - 4) / y(y - 1). Combine the terms by expanding and simplifying. It results: 3y²(y - 1) / y(y - 1) + 3y - 3 / y(y - 1) - 2y - 4 / y(y - 1). Group like terms to reduce calculation errors. The expression becomes 3y³ - 3y² + 3y - 3 - 2y + 4 / y(y - 1). Group terms and simplify 3y³ - 3y² + y + 1 / y(y - 1). This expression cannot be simplified further. Keep an eye on the details; it helps avoid silly mistakes. Always double-check each step!

Summary of Steps for 2) 3y + 3 / y - 2 × y - 2 / y² - y

  1. Clarify with parentheses: 3y + (3 / y) - 2 × (y - 2) / (y² - y)
  2. Factor Denominator: y² - y becomes y(y - 1)
  3. Complex Fraction: 3y + (3 / y) - 2 × (y - 2) / y(y - 1)
  4. Multiply: 2 × (y - 2) becomes (2y - 4) and the expression looks like 3y + (3 / y) - (2y - 4) / y(y - 1)
  5. Common Denominator: The common denominator is y(y - 1)
  6. Simplify: 3y³ - 3y² + y + 1 / y(y - 1)

Expression 3: Simplify 3) (x / x - y + x / y) × xy / x

Alright, let's get to the last one! This is (x / x - y + x / y) × xy / x. This looks like a fun one, right? Don't be intimidated; we'll break it down just like the others. This problem involves addition and multiplication, so let's remember the order of operations. Focus on getting a common denominator first when adding fractions. The parentheses indicate we need to simplify the terms inside before multiplying by xy / x. Remember, the goal is to simplify the expression to its simplest form, making it easy to understand and use. Let's do this thing!

Let's deal with the expression inside the parentheses first: x / (x - y) + x / y. We need to find a common denominator, which in this case will be y(x - y). Multiply the first fraction by y/y and the second fraction by (x-y)/(x-y). The expression becomes xy / y(x - y) + x(x - y) / y(x - y). Simplify and expand. Combine the numerators over the common denominator. We get xy + x² - xy / y(x - y). Combine like terms to simplify. The xy and -xy cancel out. We're left with x² / y(x - y). Now we've simplified the expression inside the parentheses! Now, let's tackle the multiplication part: (x² / y(x - y)) × xy / x. Before we multiply, let's simplify xy / x. The x in the numerator and the x in the denominator cancel, which gives us y. Hence, the expression becomes x² / y(x - y) × y. Multiply the fractions: x² * y / y(x - y). Now, we see that the y in the numerator and the y in the denominator cancel out, which leaves us with x² / (x - y). And there you have it! This looks a lot simpler than what we started with, right? This step really brings the whole thing home. Remember, the ability to simplify algebraic expressions is a fundamental skill in mathematics. Keep practicing, and you'll get more comfortable with them. Never be afraid to rewrite the problem to make it clearer and more understandable. The more practice you get, the quicker and more confident you'll become. So go out there and show those algebra problems who's boss!

Summary of Steps for 3) (x / x - y + x / y) × xy / x

  1. Simplify Parentheses (Common Denominator): x / (x - y) + x / y -> xy / y(x - y) + x(x - y) / y(x - y)
  2. Combine: xy + x² - xy / y(x - y) -> x² / y(x - y)
  3. Simplify: xy / x -> y
  4. Multiply: (x² / y(x - y)) * y
  5. Cancel and Simplify: x² / (x - y)