Solving 2 1/2 Y - 5 1/3 = 1 1/2 + 3/4 Y A Step-by-Step Guide

Hey guys! Today, we're diving into a fun little math problem that might seem a bit intimidating at first, but trust me, it's totally manageable once we break it down. We're going to tackle the equation 2 1/2 y - 5 1/3 = 1 1/2 + 3/4 y. This type of equation involves fractions and a variable, but don't worry, we'll go through each step together, making sure you understand exactly what's going on. Math can be super rewarding when you conquer a challenging problem, and this is one of those moments! So, let's get started and see how we can solve this equation. Our main goal here is to isolate the variable 'y' on one side of the equation. This means we need to get all the 'y' terms together and all the constant terms together. To do this, we'll use a few key mathematical principles, like adding or subtracting the same value from both sides of the equation and multiplying or dividing both sides by the same value. Remember, whatever we do to one side, we must do to the other to keep the equation balanced. This is a fundamental concept in algebra, and it's essential for solving equations like this one. We'll also be working with mixed numbers and fractions, so it's a good opportunity to brush up on those skills as well. We'll convert mixed numbers to improper fractions, find common denominators, and simplify our results. This might sound like a lot, but I promise it's not as scary as it seems. By the end of this guide, you'll have a clear understanding of how to solve this equation and similar ones. So, grab your pencils and paper, and let's embark on this mathematical adventure together! We're going to break down each step, explain the reasoning behind it, and make sure you feel confident in your ability to solve this equation. Let's get started!

Step 1: Convert Mixed Numbers to Improper Fractions

Okay, first things first, let's deal with those mixed numbers. Mixed numbers can be a bit tricky to work with directly in equations, so we're going to convert them into improper fractions. This will make our calculations much smoother and easier to manage. Remember, an improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). To convert a mixed number to an improper fraction, we use a simple process: we multiply the whole number part by the denominator, add the numerator, and then put that result over the original denominator. Let's apply this to our equation, 2 1/2 y - 5 1/3 = 1 1/2 + 3/4 y. We have three mixed numbers here: 2 1/2, 5 1/3, and 1 1/2. Let's convert them one by one. First, let's convert 2 1/2. We multiply the whole number 2 by the denominator 2, which gives us 4. Then, we add the numerator 1, which gives us 5. So, 2 1/2 becomes 5/2. Next, let's convert 5 1/3. We multiply the whole number 5 by the denominator 3, which gives us 15. Then, we add the numerator 1, which gives us 16. So, 5 1/3 becomes 16/3. Finally, let's convert 1 1/2. We multiply the whole number 1 by the denominator 2, which gives us 2. Then, we add the numerator 1, which gives us 3. So, 1 1/2 becomes 3/2. Now that we've converted all the mixed numbers to improper fractions, our equation looks like this: (5/2)y - 16/3 = 3/2 + (3/4)y. See how much cleaner that looks already? Working with improper fractions might seem a bit daunting at first, but it's a crucial skill for solving equations like this. By converting the mixed numbers, we've made the equation easier to manipulate and solve. This step is all about setting ourselves up for success in the subsequent steps. So, let's move on to the next step, where we'll start grouping like terms together and further simplifying the equation. We're making great progress, guys!

Step 2: Group Like Terms

Alright, now that we've got our equation with improper fractions, it's time to group the like terms together. This means we want to get all the terms with 'y' on one side of the equation and all the constant terms (the numbers without 'y') on the other side. This step is crucial for isolating 'y' and eventually solving for its value. Our equation currently looks like this: (5/2)y - 16/3 = 3/2 + (3/4)y. To group the 'y' terms, we can subtract (3/4)y from both sides of the equation. This will move the (3/4)y term from the right side to the left side. Remember, whatever we do to one side of the equation, we must do to the other to keep it balanced. So, we subtract (3/4)y from both sides: (5/2)y - (3/4)y - 16/3 = 3/2 + (3/4)y - (3/4)y. This simplifies to (5/2)y - (3/4)y - 16/3 = 3/2. Now, let's group the constant terms. We can add 16/3 to both sides of the equation to move the -16/3 term from the left side to the right side: (5/2)y - (3/4)y - 16/3 + 16/3 = 3/2 + 16/3. This simplifies to (5/2)y - (3/4)y = 3/2 + 16/3. Great! Now we have all the 'y' terms on the left side and all the constant terms on the right side. The next step is to combine these like terms. To do this, we'll need to find common denominators for the fractions. This will allow us to add and subtract them effectively. Grouping like terms is a fundamental technique in algebra. It helps us organize the equation and make it easier to solve. By moving the 'y' terms and the constant terms to their respective sides, we've set ourselves up for the next step, which is to simplify each side of the equation. So, let's move on to the next step and see how we can combine these fractions. We're doing awesome, guys!

Step 3: Combine Like Terms with Common Denominators

Okay, we've successfully grouped our like terms, and now it's time to combine them. This involves adding and subtracting fractions, which means we need to find common denominators. Remember, we can only add or subtract fractions if they have the same denominator. Our equation at this point is (5/2)y - (3/4)y = 3/2 + 16/3. Let's start with the left side of the equation, where we have the 'y' terms: (5/2)y - (3/4)y. The denominators here are 2 and 4. The least common multiple (LCM) of 2 and 4 is 4. So, we need to convert 5/2 to an equivalent fraction with a denominator of 4. To do this, we multiply both the numerator and the denominator of 5/2 by 2: (5 * 2) / (2 * 2) = 10/4. Now we can rewrite the left side as (10/4)y - (3/4)y. Subtracting these fractions is straightforward since they have the same denominator: (10/4)y - (3/4)y = (10 - 3)/4 y = (7/4)y. So, the left side of the equation simplifies to (7/4)y. Now, let's move on to the right side of the equation, where we have the constant terms: 3/2 + 16/3. The denominators here are 2 and 3. The least common multiple (LCM) of 2 and 3 is 6. So, we need to convert both fractions to equivalent fractions with a denominator of 6. To convert 3/2 to a fraction with a denominator of 6, we multiply both the numerator and the denominator by 3: (3 * 3) / (2 * 3) = 9/6. To convert 16/3 to a fraction with a denominator of 6, we multiply both the numerator and the denominator by 2: (16 * 2) / (3 * 2) = 32/6. Now we can rewrite the right side as 9/6 + 32/6. Adding these fractions is straightforward since they have the same denominator: 9/6 + 32/6 = (9 + 32)/6 = 41/6. So, the right side of the equation simplifies to 41/6. Our equation now looks like this: (7/4)y = 41/6. We've made significant progress in simplifying the equation. By finding common denominators and combining like terms, we've reduced the equation to a much more manageable form. The next step is to isolate 'y' completely, which we'll do by multiplying both sides of the equation by the reciprocal of the coefficient of 'y'. Let's move on to the next step and see how we can finally solve for 'y'. You guys are doing fantastic!

Step 4: Isolate 'y' by Multiplying by the Reciprocal

We're in the home stretch now! We've simplified our equation to (7/4)y = 41/6. Our final goal is to isolate 'y', which means getting 'y' all by itself on one side of the equation. To do this, we need to get rid of the fraction that's multiplying 'y', which is 7/4. The easiest way to do this is to multiply both sides of the equation by the reciprocal of 7/4. Remember, the reciprocal of a fraction is simply flipping the numerator and the denominator. So, the reciprocal of 7/4 is 4/7. Let's multiply both sides of the equation by 4/7: (4/7) * (7/4)y = (4/7) * (41/6). On the left side, (4/7) * (7/4) simplifies to 1, so we're left with just 'y': y = (4/7) * (41/6). Now, let's simplify the right side. We're multiplying two fractions, so we multiply the numerators together and the denominators together: y = (4 * 41) / (7 * 6). This gives us y = 164/42. We can simplify this fraction by finding the greatest common divisor (GCD) of 164 and 42. Both 164 and 42 are divisible by 2, so let's divide both the numerator and the denominator by 2: y = (164 / 2) / (42 / 2) = 82/21. Now, let's see if we can simplify this fraction further. The prime factorization of 82 is 2 * 41, and the prime factorization of 21 is 3 * 7. Since they don't share any common factors other than 1, the fraction 82/21 is in its simplest form. So, the solution to our equation is y = 82/21. We can also express this improper fraction as a mixed number. To do this, we divide 82 by 21. 21 goes into 82 three times (3 * 21 = 63), with a remainder of 19. So, 82/21 is equal to 3 19/21. Therefore, the solution can also be written as y = 3 19/21. We did it! We've successfully isolated 'y' and found its value. By multiplying by the reciprocal, we were able to cancel out the fraction multiplying 'y' and solve for 'y'. This step is a crucial technique in solving algebraic equations, and you've now mastered it. Let's move on to the final step, which is to verify our solution to make sure we didn't make any mistakes along the way. You guys are doing amazing!

Step 5: Verify the Solution

Awesome job, guys! We've found a solution for 'y', which is y = 82/21 (or 3 19/21). But before we celebrate too much, it's always a good idea to verify our solution. This means we're going to plug our solution back into the original equation and see if it makes the equation true. This step is crucial for catching any potential errors we might have made along the way. Our original equation was 2 1/2 y - 5 1/3 = 1 1/2 + 3/4 y. Let's substitute y = 82/21 into this equation. Remember, we converted the mixed numbers to improper fractions earlier, so let's use those forms as well: (5/2)y - 16/3 = 3/2 + (3/4)y. Now, let's plug in y = 82/21: (5/2)(82/21) - 16/3 = 3/2 + (3/4)(82/21). Let's simplify each side of the equation separately. On the left side, we have (5/2)(82/21) - 16/3. First, let's multiply (5/2)(82/21): (5 * 82) / (2 * 21) = 410/42. We can simplify this fraction by dividing both the numerator and the denominator by 2: 410/42 = 205/21. Now we have 205/21 - 16/3. To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 21 and 3 is 21. So, we need to convert 16/3 to an equivalent fraction with a denominator of 21. To do this, we multiply both the numerator and the denominator by 7: (16 * 7) / (3 * 7) = 112/21. Now we can subtract the fractions: 205/21 - 112/21 = (205 - 112)/21 = 93/21. We can simplify this fraction by dividing both the numerator and the denominator by 3: 93/21 = 31/7. So, the left side of the equation simplifies to 31/7. Now, let's simplify the right side of the equation: 3/2 + (3/4)(82/21). First, let's multiply (3/4)(82/21): (3 * 82) / (4 * 21) = 246/84. We can simplify this fraction by dividing both the numerator and the denominator by 6: 246/84 = 41/14. Now we have 3/2 + 41/14. To add these fractions, we need a common denominator. The least common multiple (LCM) of 2 and 14 is 14. So, we need to convert 3/2 to an equivalent fraction with a denominator of 14. To do this, we multiply both the numerator and the denominator by 7: (3 * 7) / (2 * 7) = 21/14. Now we can add the fractions: 21/14 + 41/14 = (21 + 41)/14 = 62/14. We can simplify this fraction by dividing both the numerator and the denominator by 2: 62/14 = 31/7. So, the right side of the equation simplifies to 31/7. We found that the left side of the equation is 31/7 and the right side of the equation is also 31/7. Since both sides are equal, our solution y = 82/21 is correct! Verifying our solution is a crucial step in the problem-solving process. It gives us confidence that our answer is accurate and that we haven't made any errors along the way. You guys have done an incredible job working through this problem! You've successfully solved a challenging equation involving fractions and a variable. You've converted mixed numbers to improper fractions, grouped like terms, found common denominators, isolated 'y', and verified your solution. You've demonstrated a strong understanding of algebraic principles, and you should be proud of your accomplishment. Keep up the great work, and you'll continue to excel in mathematics!

So there you have it! We've successfully navigated the equation 2 1/2 y - 5 1/3 = 1 1/2 + 3/4 y from start to finish. We converted mixed numbers, grouped like terms, found common denominators, isolated 'y', and even verified our solution. This journey wasn't just about finding the answer; it was about understanding the process and the why behind each step. You've equipped yourselves with valuable skills that will serve you well in future math challenges. Remember, practice makes perfect, so keep tackling these kinds of problems, and you'll become even more confident in your abilities. You guys are awesome!