Algebraic Expression Simplification: A Step-by-Step Guide

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Hey guys! Today, we're diving deep into the nitty-gritty of algebra, specifically tackling a problem that might look a little intimidating at first glance: simplifying the algebraic expression 8 1/15a + 3 2/3a - 20 4/5a. Don't sweat it, though! We're going to break this down piece by piece, making it super easy to understand. Algebra can be a blast when you get the hang of it, and mastering these kinds of problems is a huge step. We'll cover everything from understanding mixed numbers to finding common denominators and combining like terms. So, grab your notebooks, maybe a snack, and let's get this algebra party started! This isn't just about solving one problem; it's about building a solid foundation for all sorts of algebraic adventures you'll encounter down the road. Think of this as your personal algebra workout, designed to strengthen those mathematical muscles. We’ll be using bold and italics to highlight key concepts, so pay close attention! By the end of this, you'll be a pro at handling expressions with mixed numbers and variables. It’s all about patience and a systematic approach. We'll make sure to explain why we do each step, not just what we do. This way, you’ll truly understand the magic behind the math. Get ready to conquer this expression and boost your algebra confidence. Let's get those thinking caps on and make some algebraic magic happen! Remember, practice makes perfect, and we're going to practice together right now. So, no more hesitations, let's dive into the core of the problem and make it as clear as possible for everyone. We'll ensure that every step is elaborated upon, so you can follow along with ease. It’s about demystifying algebra, one problem at a time. Let’s tackle this together, and you’ll see just how manageable these kinds of problems can be. The goal is to not just solve, but to understand the process, empowering you for future challenges. So, let's get to it and make this algebraic expression our new best friend. We are going to make this super simple and fun. No complicated jargon, just clear, straightforward explanations. Your journey to algebraic mastery starts here, with this one expression. Let's get started and make some noise in the world of math!

Understanding Mixed Numbers and Variables

Understanding Mixed Numbers and Variables

Alright team, before we can even think about combining terms, we need to get a handle on what we're working with. Our expression, 8 1/15a + 3 2/3a - 20 4/5a, is made up of a few key components. First off, we have the variables. See that 'a' hanging out with each number? That's our variable. In algebra, a variable is like a placeholder for a number that we don't know or that can change. Think of it as a mystery box; we're trying to figure out what's inside, or in this case, how many of these 'a's we have in total. Since all our terms have the same variable 'a' raised to the same power (which is just 1, since there's no exponent shown), these are called like terms. This is crucial, guys, because you can only add or subtract like terms. You can't just go around adding apples and oranges, right? It's the same in algebra. We can only combine the 'a' terms.

Now, let's talk about those numbers attached to the 'a's. They're not just simple whole numbers; they're mixed numbers. A mixed number, like 8 1/15, has a whole number part (8) and a fraction part (1/15). To make our lives easier when adding and subtracting, it's usually best to convert these mixed numbers into improper fractions. An improper fraction is one where the numerator (the top number) is greater than or equal to the denominator (the bottom number). To convert a mixed number like 8 1/15 into an improper fraction, you multiply the whole number by the denominator and then add the numerator. So, for 8 1/15, it would be (8 * 15) + 1. That gives us 120 + 1, which equals 121. The denominator stays the same, so 8 1/15 becomes 121/15. We'll do this for all the mixed numbers in our expression. This conversion is a fundamental skill in algebra and arithmetic, and once you master it, you'll find many fraction-based problems become much more straightforward. It's like unlocking a new level in a game!

Let's convert the other mixed numbers too:

  • 3 2/3: (3 * 3) + 2 = 9 + 2 = 11/3. So, 3 2/3a becomes 11/3a.
  • 20 4/5: (20 * 5) + 4 = 100 + 4 = 104/5. So, 20 4/5a becomes 104/5a.

Our expression now looks like this: 121/15a + 11/3a - 104/5a. See? It's already looking a bit more manageable, right? We've taken a potentially confusing expression with mixed numbers and transformed it into an expression with improper fractions. This is a critical first step in simplifying any algebraic expression involving mixed numbers. It standardizes the format, making subsequent operations like finding common denominators and combining terms much smoother. Remember this process: multiply the whole number by the denominator, add the numerator, and keep the original denominator. Practice this a few times with different mixed numbers, and it will become second nature. It's all about building those foundational skills that make the more complex parts of algebra accessible and less daunting. So, keep this in mind as we move on to the next crucial step: finding a common denominator!

Finding a Common Denominator: The Key to Combining Fractions

Now that we've converted our mixed numbers into improper fractions, our expression is 121/15a + 11/3a - 104/5a. The next big hurdle, guys, is that we can't directly add or subtract fractions unless they have the same denominator. Think of it like trying to compare apples and pears – you need a common unit. In our case, the denominators are 15, 3, and 5. We need to find a common denominator for these three numbers. The best common denominator to find is the Least Common Multiple (LCM). This is the smallest number that all our denominators (15, 3, and 5) can divide into evenly.

Let's find the LCM of 15, 3, and 5.

  • Multiples of 15: 15, 30, 45, ...
  • Multiples of 3: 3, 6, 9, 12, 15, 18, ...
  • Multiples of 5: 5, 10, 15, 20, ...

See that? The smallest number that appears in all three lists is 15. So, our Least Common Multiple (LCM) is 15. This means 15 will be our common denominator for all the fractions in our expression. Now, we need to adjust each fraction so that it has a denominator of 15. Remember, whatever you do to the denominator, you must do to the numerator to keep the fraction's value the same. It's like multiplying the fraction by 1 (e.g., 5/5 or 3/3), which doesn't change its value.

Let's transform each fraction:

  1. For 121/15a: The denominator is already 15, so we don't need to do anything! This fraction is good to go. It stays as 121/15a.

  2. For 11/3a: Our denominator is 3, and we want it to be 15. What do we multiply 3 by to get 15? That's right, 5 (since 3 * 5 = 15). So, we multiply both the numerator and the denominator by 5: (11 * 5) / (3 * 5) = 55/15. Our term 11/3a becomes 55/15a.

  3. For 104/5a: Our denominator is 5, and we want it to be 15. What do we multiply 5 by to get 15? You guessed it, 3 (since 5 * 3 = 15). So, we multiply both the numerator and the denominator by 3: (104 * 3) / (5 * 3) = 312/15. Our term 104/5a becomes 312/15a.

Now, let's rewrite our entire expression with these new, equivalent fractions, all having the common denominator of 15:

121/15a + 55/15a - 312/15a

See how much cleaner that looks? Having a common denominator is the absolute game-changer here. It allows us to treat the numerators as if they are just coefficients of a single fraction. This step is super important in all fraction arithmetic, and especially in algebra where you're constantly manipulating expressions. Finding the LCM might seem like a small extra step, but it saves you a ton of headaches later on. It ensures accuracy and makes the process of combining terms much more straightforward. If you ever get stuck on finding the LCM, remember to list out the multiples of each number until you find the smallest one that's common to all. With a little practice, you'll be spotting LCMs like a pro. We've successfully navigated the tricky waters of common denominators, and now we're ready for the grand finale: combining those like terms!

Combining Like Terms: The Final Frontier

We've reached the home stretch, everyone! Our expression is now 121/15a + 55/15a - 312/15a. Because all the terms now have the same variable ('a') and the same denominator (15), they are officially like terms that we can combine. Combining like terms means we simply perform the operations (addition and subtraction) on the numerators, while the denominator and the variable stay the same. It's like saying we have 121 apples, we add 55 apples, and then we take away 312 apples. We're still talking about apples in the end.

So, let's focus on the numerators: 121 + 55 - 312.

First, let's add the positive numbers:

  • 121 + 55 = 176.

Now, our operation becomes: 176 - 312.

When we subtract a larger number from a smaller number, the result will be negative. Let's calculate the difference between 312 and 176:

  • 312 - 176 = 136.

Since we are doing 176 - 312, the result is negative. So, 176 - 312 = -136.

Now, we take this result and put it back with our common denominator and our variable. Our simplified expression is -136/15a.

And there you have it! We've successfully simplified the original expression 8 1/15a + 3 2/3a - 20 4/5a down to -136/15a.

It's important to note that -136/15 is an improper fraction. If your teacher or the problem asks for the answer as a mixed number, you would convert this improper fraction back. To do that, you divide the numerator (136) by the denominator (15).

  • 136 ÷ 15 = 9 with a remainder of 1.

So, 136/15 is equal to 9 and 1/15. Since our result was negative, the mixed number form would be -9 1/15a.

Both -136/15a and -9 1/15a are correct simplified forms of the original expression. The choice often depends on the specific instructions given for the problem. Typically, improper fractions are preferred in algebra because they are easier to work with in subsequent calculations.

This whole process – converting mixed numbers to improper fractions, finding a common denominator, and then combining the numerators – is a fundamental skill in algebra. It might seem like a lot of steps, but each step builds on the last, and with practice, it becomes second nature. You guys crushed it! You took a complex-looking expression and systematically broke it down into a simple, elegant answer. Remember these steps:

  1. Convert mixed numbers to improper fractions.
  2. Find the Least Common Multiple (LCM) of the denominators.
  3. Rewrite each fraction with the common denominator.
  4. Combine the numerators, keeping the denominator and variable the same.
  5. (Optional) Convert back to a mixed number if required.

By following these steps, you can tackle any similar algebraic expression with confidence. Keep practicing, and you'll become an algebra whiz in no time! High fives all around for mastering this!