Simplifying (-2x * 13 - 71) / -5x A Step-by-Step Guide

Introduction: Understanding Algebraic Expressions

Hey guys! Let's dive into simplifying algebraic expressions, a fundamental skill in mathematics. In this article, we're going to break down the expression (-2x * 13 - 71) / -5x step by step, making it super easy to understand. Simplifying expressions is like tidying up a messy room – you're making it neater and easier to work with. This process involves combining like terms, distributing, and performing operations to reduce the expression to its simplest form. Remember, mathematical expressions are the language of the universe, and simplifying them is like learning to speak that language fluently. It's not just about getting the right answer; it's about understanding the process and building a solid foundation for more advanced math. So, grab your thinking caps, and let's get started!

Breaking Down the Numerator: -2x * 13 - 71

First, let's tackle the numerator: -2x * 13 - 71. To simplify this, we need to perform the multiplication first, following the order of operations (PEMDAS/BODMAS). When we multiply -2x by 13, we get -26x. So now, our numerator looks like this: -26x - 71. Can we simplify this further? Nope! -26x and -71 are unlike terms, meaning one has a variable (x) and the other is just a constant. We can't combine them, just like you can't mix apples and oranges. This part of the expression is now as simple as it gets. Remember, simplifying is like mathematical decluttering – we're getting rid of the unnecessary stuff and making it easier to see what's important. This step is crucial because it lays the groundwork for the rest of the simplification process. Without a properly simplified numerator, the subsequent steps will be more challenging, and the final result may be incorrect. So, let’s take a moment to appreciate the simplicity we’ve achieved so far. We’ve transformed a seemingly complex part of the expression into something much more manageable. This is the essence of simplifying: breaking down a big problem into smaller, easier-to-handle pieces.

Dealing with the Denominator: -5x

Now, let's shift our focus to the denominator: -5x. Guess what? It's already in its simplest form! There's nothing to combine or simplify here. Sometimes, the simplest things are the best, right? The denominator, -5x, represents a term with a variable, and it's ready to go. This might seem like a small step, but it's an important reminder that not everything needs to be complicated. We've got our simplified numerator and our simplified denominator, and now we're ready to put them together. Understanding the denominator is just as crucial as understanding the numerator. The denominator tells us what we're dividing by, and it plays a vital role in the overall value of the expression. In this case, the denominator is a single term, which makes our job a bit easier. However, in more complex expressions, the denominator might require its own simplification process. So, always pay close attention to both the numerator and the denominator – they work together to determine the value of the expression. With the denominator sorted, we're one step closer to the final simplified form. It's like having all the ingredients ready for a recipe – now we just need to put them together.

Putting It All Together: (-26x - 71) / -5x

Okay, we've simplified the numerator and the denominator separately. Now comes the exciting part: putting it all together! Our expression now looks like this: (-26x - 71) / -5x. Can we simplify this fraction further? Well, we can't cancel out any terms directly because the numerator has two terms separated by a subtraction, and we can only cancel out factors that are common to the entire numerator and the entire denominator. However, we can do something pretty cool: we can split the fraction into two separate fractions. This is like dividing a pizza into slices – we're just looking at the expression in a slightly different way. So, let's rewrite the expression as (-26x / -5x) + (-71 / -5x). See what we did there? Now we have two fractions, and we can simplify each one individually. This technique is super useful in algebra, so keep it in your mathematical toolbox! Remember, the goal of simplifying is to make the expression as clear and concise as possible. By splitting the fraction, we've opened up new avenues for simplification and brought ourselves closer to the final answer. This step demonstrates a key principle in problem-solving: sometimes, breaking a problem down into smaller parts makes it easier to solve. So, let's continue our journey and see what simplifications we can make in these new fractions.

Simplifying the Split Fractions: -26x / -5x and -71 / -5x

Let's simplify each fraction we got from the split: -26x / -5x and -71 / -5x. For the first fraction, -26x / -5x, we can see that both the numerator and the denominator have 'x' as a common factor. Also, both are negative, and a negative divided by a negative is a positive. So, the 'x's cancel out, and the negatives cancel out, leaving us with 26/5. That's a neat simplification, right? Now, let's look at the second fraction: -71 / -5x. Again, we have a negative divided by a negative, which will give us a positive. So, we can rewrite this as 71 / 5x. There's nothing else to simplify here because 71 and 5 don't have any common factors, and the 'x' in the denominator stays put. We've successfully simplified both fractions! This part of the process highlights the power of recognizing common factors and applying the rules of division. Simplifying fractions is a crucial skill in algebra, and it's something you'll use again and again. By carefully examining each fraction, we were able to eliminate common terms and reduce the expression to its simplest form. This step is like polishing a gemstone – we're revealing the beauty and clarity that were hidden within the original expression. So, let's take a moment to appreciate the progress we've made and prepare to combine these simplified fractions into our final answer.

The Grand Finale: Combining the Simplified Fractions

We're almost there, guys! We've simplified -26x / -5x to 26/5 and -71 / -5x to 71 / 5x. Now, let's put these simplified fractions back together. Our expression now looks like this: 26/5 + 71 / 5x. This is the simplified form of the original expression! We can't combine these two terms any further because they have different denominators (one has 'x' in the denominator, and the other doesn't). So, we've reached the end of our simplification journey. Give yourselves a pat on the back! You've taken a complex expression and broken it down into its simplest components. This final step is like reaching the summit of a mountain – we've overcome all the challenges and arrived at our destination. The simplified expression, 26/5 + 71 / 5x, is much easier to work with than the original. It clearly shows the relationship between the variables and constants, and it's in a form that we can easily use for further calculations or analysis. This is the power of simplification: it transforms complex expressions into manageable forms, making mathematics more accessible and enjoyable. So, let's celebrate our success and remember the valuable skills we've learned along the way.

Conclusion: The Power of Simplification

Simplifying algebraic expressions might seem daunting at first, but as we've seen, it's all about breaking things down step by step. We started with a seemingly complex expression, (-2x * 13 - 71) / -5x, and through careful simplification, we arrived at 26/5 + 71 / 5x. Remember, the key is to follow the order of operations, combine like terms, and don't be afraid to split fractions if it helps! Simplifying expressions is a fundamental skill in mathematics, and it's one that will serve you well in all your future math endeavors. It's not just about getting the right answer; it's about understanding the process and developing your problem-solving skills. So, keep practicing, keep exploring, and most importantly, keep simplifying! You've got this!

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