Solving (7x+4)/(8y) - (3x-1)/(6y) A Step By Step Guide

Hey guys! Ever feel like fraction operations are just a jumbled mess of numerators and denominators? Well, you're not alone! But fear not, because today, we're going to break down a seemingly complex problem into manageable steps. We'll be tackling the expression (7x+4)/(8y) - (3x-1)/(6y), which involves subtracting fractions with algebraic expressions. Sounds intimidating? Don't worry, we'll make it crystal clear.

Understanding the Fundamentals of Fraction Subtraction

Before we dive into the specifics of our problem, let's quickly recap the basics of fraction subtraction. Remember, you can only subtract fractions if they have a common denominator. This means the bottom numbers (denominators) of the fractions must be the same. If they're not, we need to find the least common multiple (LCM) of the denominators and adjust the fractions accordingly. Think of it like this: you can't directly compare or subtract apples and oranges – you need a common unit, like "fruit." Similarly, fractions need a common denominator to be subtracted.

To find the LCM, we look for the smallest number that both denominators divide into evenly. Once we have the LCM, we multiply the numerator and denominator of each fraction by a factor that will make the denominator equal to the LCM. This process ensures that we're working with equivalent fractions, meaning we haven't changed the value of the original fractions, just their appearance. For example, if we want to subtract 1/2 and 1/3, the LCM of 2 and 3 is 6. So, we multiply the numerator and denominator of 1/2 by 3 to get 3/6, and we multiply the numerator and denominator of 1/3 by 2 to get 2/6. Now we can easily subtract: 3/6 - 2/6 = 1/6.

Understanding this fundamental concept of common denominators is crucial for tackling algebraic fractions. The same principles apply, but instead of just numbers, we're dealing with expressions involving variables. This might seem a bit trickier, but with a systematic approach, it becomes much simpler. We'll apply these principles to our specific problem, (7x+4)/(8y) - (3x-1)/(6y), in the next section.

Finding the Least Common Denominator (LCD)

Okay, let's get our hands dirty with our problem: (7x+4)/(8y) - (3x-1)/(6y). The first hurdle is finding the least common denominator (LCD) of 8y and 6y. Remember, the LCD is the smallest expression that both 8y and 6y divide into evenly. To find it, we need to consider both the numerical coefficients (8 and 6) and the variable part (y).

Let's start with the numerical coefficients. What's the least common multiple (LCM) of 8 and 6? Think about the multiples of each number: Multiples of 8: 8, 16, 24, 32, ... Multiples of 6: 6, 12, 18, 24, 30, ... The LCM of 8 and 6 is 24. That's the numerical part of our LCD.

Now, let's look at the variable part. Both denominators have 'y' in them. Since we need the smallest expression that both divide into, we only need 'y' in our LCD. We don't need y² or any higher power of y, because 8y and 6y already contain 'y' to the power of 1.

Putting it all together, the LCD of 8y and 6y is 24y. This is the common denominator we'll use to rewrite our fractions so that we can subtract them. This might seem like a small step, but it's a critical one. Finding the correct LCD is the foundation for successfully subtracting fractions, especially when dealing with algebraic expressions. Once we have the LCD, we can move on to the next step: rewriting the fractions with this common denominator.

Rewriting Fractions with the LCD

Now that we've found the LCD, 24y, we need to rewrite our fractions, (7x+4)/(8y) and (3x-1)/(6y), so they both have this denominator. Remember, the key is to multiply both the numerator and denominator of each fraction by the same factor. This ensures that we're creating equivalent fractions, which means we're not changing the value of the fractions, just their appearance.

Let's start with the first fraction, (7x+4)/(8y). We need to figure out what to multiply 8y by to get 24y. If you're not sure, you can divide 24y by 8y, which gives you 3. So, we need to multiply both the numerator and denominator of the first fraction by 3: 3 * (7x + 4) / 3 * (8y) = (21x + 12) / 24y Notice how we distributed the 3 to both terms in the numerator (7x and 4). This is a crucial step in working with algebraic fractions. Make sure you multiply the factor by every term in the numerator.

Now, let's move on to the second fraction, (3x-1)/(6y). We need to figure out what to multiply 6y by to get 24y. Again, we can divide 24y by 6y, which gives us 4. So, we need to multiply both the numerator and denominator of the second fraction by 4: 4 * (3x - 1) / 4 * (6y) = (12x - 4) / 24y Again, we distributed the 4 to both terms in the numerator (3x and -1). It's really important to pay attention to the signs here. Multiplying -1 by 4 gives us -4.

So, we've successfully rewritten our fractions with the LCD: (7x+4)/(8y) has become (21x + 12) / 24y, and (3x-1)/(6y) has become (12x - 4) / 24y. Now we can finally subtract them, which is the next step in our journey to mastering fraction operations! This is where things really start to come together.

Performing the Subtraction

Alright, we've done the hard work of finding the LCD and rewriting the fractions. Now comes the satisfying part: subtracting them! We have (21x + 12) / 24y - (12x - 4) / 24y. Remember, when subtracting fractions with a common denominator, we simply subtract the numerators and keep the denominator the same.

So, we need to subtract (12x - 4) from (21x + 12). This is where it's super important to be careful with the signs. We're essentially distributing a negative sign to the second numerator: (21x + 12) - (12x - 4) = 21x + 12 - 12x + 4 Notice how the -4 became +4 because we were subtracting a negative number. This is a common mistake, so double-check your signs! Now, we can combine like terms: 21x - 12x = 9x 12 + 4 = 16 So, the numerator becomes 9x + 16.

Therefore, our subtraction result is (9x + 16) / 24y. We've successfully subtracted the fractions! But before we declare victory, there's one more important step: simplifying the result. This is like the final polish on a beautiful piece of work. We want to make sure our answer is in its simplest form. Let's see if we can simplify our fraction in the next section.

Simplifying the Result

We've arrived at (9x + 16) / 24y after subtracting the fractions. The final step is to simplify the result, if possible. Simplifying a fraction means reducing it to its lowest terms. This involves looking for common factors in the numerator and denominator and dividing them out. Think of it like finding the greatest common divisor (GCD) and dividing both the top and bottom by it.

In our case, we have (9x + 16) / 24y. Let's examine the numerator and denominator separately. In the numerator, we have the expression 9x + 16. Is there a common factor between 9 and 16? No, the factors of 9 are 1, 3, and 9, while the factors of 16 are 1, 2, 4, 8, and 16. The only common factor is 1, which doesn't help us simplify. Furthermore, there's no 'x' term in the 16, so we can't factor out an 'x' either.

Now, let's look at the denominator, 24y. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. We already know that 9 and 16 don't share any common factors other than 1, so we don't need to worry about simplifying the numerical coefficients further. The denominator also has a 'y' term, but the numerator doesn't have a 'y', so we can't cancel out any 'y's.

Since there are no common factors between the numerator (9x + 16) and the denominator (24y) other than 1, the fraction (9x + 16) / 24y is already in its simplest form. We can't simplify it any further. This means we've reached the final answer! Woohoo!

Conclusion: Mastering Fraction Operations

So, guys, we've successfully navigated the world of fraction subtraction with algebraic expressions! We tackled the problem (7x+4)/(8y) - (3x-1)/(6y), and we broke it down into manageable steps. We found the LCD, rewrote the fractions, performed the subtraction, and simplified the result. Our final answer is (9x + 16) / 24y. Isn't it satisfying to see a complex problem become so clear when you approach it step-by-step?

The key takeaways here are: Always find the least common denominator (LCD) first. Remember to distribute when multiplying numerators. Be extra careful with signs when subtracting. And finally, always simplify your answer as much as possible. These skills are essential not just for algebra, but for many areas of mathematics and beyond.

Practice makes perfect, so try tackling similar problems. The more you practice, the more comfortable you'll become with fraction operations. You'll start to see patterns and develop an intuition for how to approach these problems. And remember, if you ever get stuck, don't hesitate to review the steps we've covered today. Keep up the great work, and you'll be a fraction master in no time!