Hey there, math enthusiasts! Ever stumbled upon an equation that looks like a jumbled mess of fractions and variables? Well, today, we're going to break down one such equation and make it crystal clear. We're diving deep into the world of algebraic fractions to tackle the sum: (3y)/(y^2+7y+10) + 2/(y+2). Buckle up, because we're about to embark on a journey of simplification and problem-solving!
Deconstructing the Algebraic Expression
At first glance, this expression might seem intimidating, but don't worry, we'll take it step by step. The key to solving this lies in understanding how to add fractions, especially when they involve algebraic expressions. The golden rule of fraction addition is that you need a common denominator. Think of it like trying to add apples and oranges – you can't directly add them until you have a common unit, like "fruits." In the case of fractions, the common unit is the denominator.
So, let's break down our expression into its components. We have two fractions here: (3y)/(y^2+7y+10) and 2/(y+2). The denominators are (y^2+7y+10) and (y+2), respectively. Our mission, should we choose to accept it (and we do!), is to find a common denominator for these two.
Now, this is where our algebraic skills come into play. We need to see if we can factor the more complex denominator, which is (y^2+7y+10). Factoring is like reverse multiplication – we're trying to find two expressions that, when multiplied together, give us the original expression. In this case, we're looking for two numbers that add up to 7 (the coefficient of the y term) and multiply to 10 (the constant term). The numbers 2 and 5 fit the bill perfectly! So, we can factor (y^2+7y+10) as (y+2)(y+5).
Aha! A breakthrough! We've discovered that one of the factors of the first denominator is the same as the denominator of the second fraction, which is (y+2). This is excellent news because it means our common denominator is within reach. The least common denominator (LCD) for these two fractions is simply (y+2)(y+5). It's the smallest expression that both denominators can divide into evenly.
Finding the Common Denominator and Rewriting Fractions
Now that we've identified the common denominator as (y+2)(y+5), we need to rewrite each fraction with this new denominator. Remember, we can't just change the denominator without changing the numerator as well, or we'll be changing the value of the fraction. It's like scaling a recipe – if you double the ingredients, you need to double everything to maintain the same flavor.
Let's start with the first fraction, (3y)/(y^2+7y+10). We already factored the denominator as (y+2)(y+5), so this fraction already has the common denominator! That was easy, right?
Now, for the second fraction, 2/(y+2), we need to multiply both the numerator and the denominator by (y+5) to get the common denominator. This gives us [2(y+5)]/[(y+2)(y+5)], which simplifies to (2y+10)/[(y+2)(y+5)]. See what we did there? We multiplied both the top and bottom by the same expression, so we're essentially multiplying by 1, which doesn't change the value of the fraction.
So, now we've rewritten our original expression as: (3y)/[(y+2)(y+5)] + (2y+10)/[(y+2)(y+5)]. Look at that! Both fractions have the same denominator. We're one giant leap closer to solving this.
Combining the Fractions and Simplifying
The moment we've been waiting for! Now that we have a common denominator, we can finally add the fractions. Adding fractions with the same denominator is a breeze – we simply add the numerators and keep the denominator the same. It's like adding slices of the same pizza – if you have 3 slices and add 2 more, you have 5 slices in total.
So, adding the numerators of our fractions, we get 3y + (2y+10), which simplifies to 5y + 10. Our combined fraction now looks like this: (5y+10)/[(y+2)(y+5)]. We're making serious progress!
But hold on, we're not quite done yet. In mathematics, we always want to simplify our expressions as much as possible. This is where our keen eyes for patterns and factors come in handy. Do you notice anything we can factor out of the numerator, (5y+10)? That's right! Both terms are divisible by 5. Factoring out a 5, we get 5(y+2).
Now, our fraction looks like this: [5(y+2)]/[(y+2)(y+5)]. And now, the magic happens! We see that we have a common factor of (y+2) in both the numerator and the denominator. We can cancel these out, just like simplifying a regular fraction by dividing both the top and bottom by a common factor.
Canceling out the (y+2) terms, we're left with our final, simplified answer: 5/(y+5). Ta-da! We've successfully navigated the world of algebraic fractions and arrived at a sleek, simplified solution.
The Grand Finale: The Simplified Sum
After our epic journey through factoring, finding common denominators, and simplifying, we've arrived at the final destination. The sum of the algebraic expression (3y)/(y^2+7y+10) + 2/(y+2), in its simplest form, is 5/(y+5). Give yourselves a pat on the back, guys! You've conquered a complex-looking equation and emerged victorious.
This whole process highlights the beauty and elegance of mathematics. What seemed like a daunting problem at first glance, with a bit of strategic thinking and step-by-step execution, transformed into a manageable and even enjoyable challenge. Remember, math isn't about memorizing formulas; it's about understanding the underlying principles and applying them creatively.
So, the next time you encounter a complex algebraic expression, don't shy away from it. Embrace the challenge, break it down into smaller steps, and unleash your inner mathematician. You might be surprised at what you can achieve! And remember, practice makes perfect. The more you work with algebraic fractions, the more comfortable and confident you'll become. Keep exploring, keep learning, and keep unlocking the secrets of the mathematical universe!
