Simplifying Algebraic Expressions A Step-by-Step Guide

Hey guys! Today, we're going to break down a common algebra problem that might seem tricky at first glance but is totally manageable with the right steps. We're talking about simplifying expressions involving fractions with variables. Specifically, we’ll tackle the expression ${\frac{5}{m}-\frac{m+8}{m^2-4 m}}$. This kind of problem often appears in algebra courses, and mastering it will definitely boost your confidence and skills. So, let's dive in and make sure we understand each step clearly. Our main goal here is to find an equivalent expression by simplifying the given one. This involves a few key techniques, including finding common denominators, factoring, and combining like terms. Don't worry if these terms sound intimidating; we'll go through each one methodically. By the end of this guide, you'll not only know the answer but also understand why it's the answer. We'll cover everything from identifying the problem type to avoiding common mistakes, ensuring you’re well-prepared for similar challenges in the future. So, grab your pencils, and let’s get started on this algebraic adventure!

Understanding the Problem

Before we jump into solving, let’s make sure we understand the expression we’re dealing with. We have ${\frac{5}{m}-\frac{m+8}{m^2-4 m}}$. Notice that this is a subtraction of two fractions. The first fraction has a simple denominator, m, while the second has a more complex denominator, m² - 4m. To subtract these fractions, we need a common denominator. This is a fundamental concept in fraction arithmetic, and it’s just as crucial in algebra. A common denominator allows us to combine the numerators, which is essential for simplifying the expression. We need to find a common multiple of m and m² - 4m. Factoring the second denominator will help us identify the least common multiple. This is where our factoring skills come into play. Factoring is like reverse distribution; we’re breaking down an expression into its multiplicative components. In this case, we’ll factor m² - 4m. Keep in mind that finding the common denominator is not just about finding any common multiple; we want the least common multiple to keep our calculations as simple as possible. So, let's move on to factoring the denominator, which is the next key step in simplifying our expression. Remember, the goal is to make the denominators of both fractions the same so we can combine them.

Factoring the Denominator

The next crucial step in simplifying our expression is factoring the denominator of the second fraction, which is m² - 4m. Factoring is a technique that allows us to rewrite an expression as a product of its factors. In this case, we're looking for common factors in the terms m² and -4m. When we look at m² - 4m, we can see that both terms have a common factor of m. So, we can factor out m from the expression. This gives us m(m - 4). Factoring the denominator like this is a game-changer because it helps us identify the least common denominator (LCD) more easily. Now that we have m² - 4m factored as m(m - 4), we can see that the denominators of our original fractions are m and m(m - 4). This makes it clear what the LCD should be. The LCD is the smallest expression that both denominators can divide into evenly. In this case, the LCD is m(m - 4). This is because the first fraction already has m in its denominator, and the second fraction has m(m - 4). Factoring the denominator not only simplifies the process of finding the LCD but also sets us up for the next step: rewriting the fractions with the common denominator. This step is vital for combining the fractions and simplifying the entire expression. So, with the denominator factored, we're one step closer to the final solution.

Finding the Common Denominator

Now that we've factored the denominator m² - 4m into m(m - 4), finding the common denominator becomes much simpler. Remember, the original expression is ${\frac{5}{m}-\frac{m+8}{m^2-4 m}}$. We've already determined that the factored form of the second denominator is m(m - 4). This means our two denominators are now m and m(m - 4). The least common denominator (LCD) is the smallest expression that both denominators can divide into without leaving a remainder. In this case, the LCD is m(m - 4). To get both fractions to have this denominator, we need to adjust the first fraction. The first fraction is ${\frac{5}{m}}$. To make its denominator m(m - 4), we need to multiply both the numerator and the denominator by (m - 4). This gives us ${\frac{5(m - 4)}{m(m - 4)}}$. The second fraction, ${\frac{m+8}{m(m-4)}}$, already has the common denominator, so we don't need to change it. Now that both fractions have the same denominator, we can proceed to the next step: combining the numerators. This is a crucial step in simplifying the expression, as it allows us to perform the subtraction and reduce the expression to its simplest form. So, let's move on and combine those numerators!

Rewriting Fractions with the Common Denominator

Alright, guys, let's talk about rewriting fractions to have that all-important common denominator. We've established that our common denominator is m(m - 4). Our original expression is ${\frac{5}{m}-\frac{m+8}{m(m-4)}}$. The first fraction, ${\frac{5}{m}}$, needs to be adjusted. To get the denominator to be m(m - 4), we need to multiply both the numerator and the denominator by (m - 4). This gives us:

${\frac{5}{m} \times \frac{m - 4}{m - 4} = \frac{5(m - 4)}{m(m - 4)}}$

Now, let's simplify the numerator of this new fraction. We distribute the 5 across (m - 4), which gives us 5m - 20. So, our first fraction now looks like this:

${\frac{5m - 20}{m(m - 4)}}$

The second fraction, ${\frac{m+8}{m(m-4)}}$, already has the common denominator, so we don’t need to change it. It stays as:

${\frac{m + 8}{m(m - 4)}}$

Now that both fractions have the same denominator, we can rewrite our original expression as:

${\frac{5m - 20}{m(m - 4)} - \frac{m + 8}{m(m - 4)}}$

This is a huge step forward! We’ve successfully rewritten the fractions with a common denominator, which means we’re ready to combine the numerators. Combining the numerators is where the magic happens, as it allows us to simplify the expression further and get closer to our final answer. So, let's move on to that next crucial step.

Combining the Numerators

Okay, time to get those numerators combined! We've got our expression rewritten with the common denominator m(m - 4):

${\frac{5m - 20}{m(m - 4)} - \frac{m + 8}{m(m - 4)}}$

Since the denominators are the same, we can now subtract the numerators. This means we’ll subtract (m + 8) from (5m - 20). It’s super important to remember to distribute the negative sign correctly. When we subtract (m + 8), it’s the same as subtracting m and subtracting 8. So, we have:

${(5m - 20) - (m + 8) = 5m - 20 - m - 8}$

Now, let’s combine the like terms. We have 5m and -m, which combine to give us 4m. And we have -20 and -8, which combine to give us -28. So, our numerator simplifies to:

${4m - 28}$

Now we put this back over our common denominator m(m - 4), and our expression looks like this:

${\frac{4m - 28}{m(m - 4)}}$

We’re not quite done yet, guys! We’ve combined the numerators, but we can still simplify this expression further. The next step is to look for any common factors in the numerator and see if we can reduce the fraction. This is where factoring comes in handy again. So, let’s move on to the next step: factoring the numerator.

Factoring the Numerator

Now that we've combined the numerators, we have the expression ${\frac{4m - 28}{m(m - 4)}}$. To simplify further, we should look for any common factors in the numerator, 4m - 28. Factoring is a powerful tool, guys, and it's super useful here. When we examine 4m - 28, we can see that both terms are divisible by 4. So, we can factor out a 4 from the numerator:

${4m - 28 = 4(m - 7)}$

This means we can rewrite our expression as:

${\frac{4(m - 7)}{m(m - 4)}}$

Now, let's take a look at our entire expression. We have a factored numerator and a factored denominator. We need to check if there are any common factors between the numerator and the denominator that we can cancel out. In this case, we have 4(m - 7) in the numerator and m(m - 4) in the denominator. There are no common factors that we can cancel out. The expression is now in its simplest form. Factoring the numerator has helped us to ensure that we've simplified the expression as much as possible. We've gone from the original expression to a much cleaner and simpler form. So, let's recap our steps and make sure we're confident with our solution.

Final Simplified Expression

Alright, let's bring it all together and see our final simplified expression. We started with ${\frac{5}{m}-\frac{m+8}{m^2-4 m}}$ and went through several key steps: understanding the problem, factoring the denominator, finding the common denominator, rewriting the fractions, combining numerators, and factoring the numerator. After all that work, we arrived at the simplified expression:

${\frac{4(m - 7)}{m(m - 4)}}$

This is our final answer! We can confidently say that this expression is equivalent to the original one, but it's in a much simpler and easier-to-work-with form. So, how did we get here? Remember, the key was to break down the problem into manageable steps. We first identified the need for a common denominator, which led us to factor the denominator m² - 4m. This made it clear that our LCD was m(m - 4). Then, we rewrote the fractions with this common denominator, combined the numerators, and simplified. Factoring the numerator was the final touch, ensuring we had the expression in its simplest form. This process illustrates the power of methodical problem-solving in algebra. By taking each step deliberately and carefully, we transformed a seemingly complex expression into something much more manageable. So, let's take a moment to appreciate our journey and the skills we’ve honed along the way!

Conclusion: Mastering Algebraic Simplification

Wrapping things up, guys, we've successfully simplified the algebraic expression ${\frac{5}{m}-\frac{m+8}{m^2-4 m}}$ and arrived at the equivalent expression ${\frac{4(m - 7)}{m(m - 4)}}$. This journey through simplification highlights several crucial algebraic techniques. We started by understanding the problem, recognizing the need for a common denominator when subtracting fractions. This led us to the important step of factoring, both the denominator and, later, the numerator. Factoring is a fundamental skill in algebra, and it allows us to break down complex expressions into simpler components. Finding the least common denominator (LCD) was another key step. The LCD allows us to rewrite fractions so that they can be combined, making the simplification process much smoother. We also emphasized the importance of careful arithmetic, particularly when distributing negative signs while combining numerators. A small mistake in arithmetic can lead to a completely wrong answer, so accuracy is paramount. Finally, we saw how simplifying an expression isn't just about getting the right answer; it’s about making the expression easier to understand and work with. A simplified expression can reveal important properties and make further calculations much more straightforward. By mastering these techniques, you'll be well-equipped to tackle a wide range of algebraic problems. Keep practicing, and you'll find these skills become second nature. Great job, everyone, on simplifying this expression! Keep up the excellent work, and you’ll conquer algebra in no time!