Simplifying Algebraic Fractions: A Step-by-Step Guide

by SLV Team 54 views

Hey math enthusiasts! Today, we're diving into the world of algebraic expressions, specifically focusing on simplifying them. This is super important because it's a fundamental skill you'll use throughout your algebra journey. We'll break down two examples, making sure you grasp the concepts and can tackle similar problems with confidence. So, let's roll up our sleeves and get started!

Example 1: Simplifying x²/(x²-9) - x/(x-3)

Let's start with our first expression: x²/(x²-9) - x/(x-3). Looks a bit intimidating at first, right? But don't worry, we'll break it down step by step. Our main goal here is to simplify the given algebraic expression and combine the fractions. The first thing we need to do is factorize the denominator of the first fraction. Notice that x² - 9 is a difference of squares, which can be factored into (x - 3)(x + 3). So, we rewrite the expression as: x²/((x-3)(x+3)) - x/(x-3). Now, we have two fractions with different denominators. To subtract them, we need a common denominator. Looking at the denominators, we see that (x - 3) is already a factor in both, but the first fraction also has (x + 3). Therefore, our common denominator will be (x - 3)(x + 3).

To get the first fraction to have this denominator, it already has it, so no changes are needed. The second fraction, x/(x-3), needs to be multiplied by (x + 3) in both the numerator and the denominator. So we get (x * (x + 3))/((x - 3)(x + 3)). Now the expression looks like this: x²/((x-3)(x+3)) - (x(x+3))/((x-3)(x+3)).

Next, we need to combine the numerators over the common denominator: (x² - x(x + 3))/((x - 3)(x + 3)). Let's expand the term x(x + 3) in the numerator, which becomes x² + 3x. Our expression now looks like this: (x² - (x² + 3x))/((x - 3)(x + 3)). Now, let's distribute the negative sign in the numerator: (x² - x² - 3x)/((x - 3)(x + 3)). Notice that the x² terms cancel out, leaving us with: -3x/((x - 3)(x + 3)).

Finally, we can try to simplify further. However, there are no common factors between the numerator and denominator. So, our simplified expression is -3x/((x - 3)(x + 3)). We can also write this as -3x/(x² - 9). And that, my friends, is how you simplify this algebraic expression! It's a combination of factoring, finding a common denominator, and careful simplification.

Step-by-Step Breakdown for Clarity

To make sure we're all on the same page, let's recap the steps we took:

  1. Factor the Denominator: Rewrite x² - 9 as (x - 3)(x + 3).
  2. Identify Common Denominator: The common denominator is (x - 3)(x + 3).
  3. Rewrite Fractions: Adjust the second fraction to have the common denominator: x/(x-3) becomes (x(x+3))/((x-3)(x+3)).
  4. Combine Numerators: Subtract the numerators: (x² - x(x + 3))/((x - 3)(x + 3)).
  5. Simplify: Expand and simplify the numerator: (x² - x² - 3x)/((x - 3)(x + 3)) which simplifies to -3x/((x - 3)(x + 3)).
  6. Final Result: The simplified form is -3x/((x - 3)(x + 3)) or -3x/(x² - 9).

Example 2: Simplifying 3y/(y²-9) + y/(3y-9)

Alright, let's tackle our second example: 3y/(y²-9) + y/(3y-9). We'll use similar techniques here to simplify this algebraic fraction. First things first, we'll need to factorize the denominators. The first denominator, y² - 9, is another difference of squares and factors into (y - 3)(y + 3). The second denominator, 3y - 9, has a common factor of 3, which we can factor out: 3(y - 3). So, the expression becomes: 3y/((y - 3)(y + 3)) + y/(3(y - 3)). Next, we need to figure out our common denominator. We see (y - 3) is a factor in both denominators. The first one has (y + 3), and the second has a 3. Thus, our common denominator will be 3(y - 3)(y + 3).

Now, we need to rewrite each fraction with this common denominator. The first fraction, 3y/((y - 3)(y + 3)), needs to be multiplied by 3 in both the numerator and the denominator. This gives us (3 * 3y)/(3(y - 3)(y + 3)), which simplifies to 9y/(3(y - 3)(y + 3)). The second fraction, y/(3(y - 3)), needs to be multiplied by (y + 3) in both the numerator and denominator. This gives us (y(y + 3))/(3(y - 3)(y + 3)), which simplifies to (y² + 3y)/(3(y - 3)(y + 3)). Our expression now looks like this: 9y/(3(y - 3)(y + 3)) + (y² + 3y)/(3(y - 3)(y + 3)).

We are now ready to combine the numerators over the common denominator: (9y + y² + 3y)/(3(y - 3)(y + 3)). Let's combine like terms in the numerator: (y² + 12y)/(3(y - 3)(y + 3)). We can factor a y out of the numerator, giving us: y(y + 12)/(3(y - 3)(y + 3)). However, we can't simplify this expression further because there are no common factors between the numerator and the denominator that can be cancelled out. Therefore, our simplified expression is y(y + 12)/(3(y - 3)(y + 3)). We could also expand the denominator, which would give us y(y + 12)/(3(y² - 9)) or (y² + 12y)/(3y² - 27). Guys, we've successfully simplified the algebraic fraction.

Step-by-Step Breakdown for Clarity

Let's revisit the steps we took:

  1. Factor the Denominators: Factor y² - 9 as (y - 3)(y + 3) and 3y - 9 as 3(y - 3).
  2. Identify Common Denominator: The common denominator is 3(y - 3)(y + 3).
  3. Rewrite Fractions: Adjust the fractions to have the common denominator.
    • 3y/((y - 3)(y + 3)) becomes 9y/(3(y - 3)(y + 3)).
    • y/(3(y - 3)) becomes (y(y + 3))/(3(y - 3)(y + 3)).
  4. Combine Numerators: Add the numerators: (9y + y² + 3y)/(3(y - 3)(y + 3)).
  5. Simplify: Combine like terms and factor: (y² + 12y)/(3(y - 3)(y + 3)) which simplifies to y(y + 12)/(3(y - 3)(y + 3)).
  6. Final Result: The simplified form is y(y + 12)/(3(y - 3)(y + 3)) or (y² + 12y)/(3y² - 27).

Key Takeaways and Tips for Simplifying Algebraic Expressions

Alright, folks, let's wrap things up with some key takeaways. Simplifying algebraic expressions might seem daunting at first, but with practice, it becomes second nature. Here are some pro tips to help you along the way:

  • Factoring is Your Friend: Always, always, start by factoring the denominators. This makes it easier to find the common denominator and simplify your expression. Remember difference of squares, common factors and other factoring techniques. The ability to factor is crucial.

  • Find the Common Denominator: The common denominator is the least common multiple (LCM) of the denominators. This is essential for adding or subtracting fractions. Make sure you understand how to find the LCM, especially with algebraic expressions.

  • Rewrite Each Fraction: Once you know the common denominator, rewrite each fraction so that it has the same denominator. This means multiplying both the numerator and denominator by whatever is needed to get the common denominator.

  • Combine and Simplify: After getting the common denominator, combine the numerators. Then, simplify the resulting expression by combining like terms and factoring if possible. Always look for ways to simplify your answer.

  • Practice, Practice, Practice: The more you practice, the better you'll become at simplifying algebraic expressions. Work through various examples, starting with easier ones and gradually moving to more complex problems. Use online resources, textbooks, and practice worksheets.

  • Understand the Rules: Make sure you have a solid understanding of the rules of exponents, the order of operations, and the different factoring techniques. These are the building blocks of simplifying algebraic expressions.

  • Check Your Work: After simplifying, double-check your answer to make sure you haven't made any mistakes. You can do this by plugging in a value for the variable and seeing if the original expression and the simplified expression give you the same result.

  • Don't Give Up: Algebra can be challenging, but don't let that discourage you. Take your time, break down problems into smaller steps, and keep practicing. You'll get there!

  • Focus on the Fundamentals: Master basic skills like adding, subtracting, multiplying, and dividing fractions, as well as working with integers and signed numbers.

Conclusion: Mastering Algebraic Fractions

So there you have it, guys! We've covered how to simplify two different algebraic expressions, and hopefully, you now have a better understanding of the process. Remember, the key is to factorize, find a common denominator, rewrite the fractions, and simplify. Algebraic expressions are a cornerstone of algebra, and mastering them is crucial for success. Don't be afraid to take your time, practice, and ask questions. With dedication, you'll be simplifying algebraic fractions like a pro in no time! Keep practicing and good luck!