Solving 31e = 2/8 + 3/8 A Step-by-Step Mathematical Guide

Hey guys! Ever stumbled upon a math problem that looks like a jumbled mess of numbers and fractions? Don't sweat it! We've all been there. Today, we're going to break down one such problem, 31e = 2/8 + 3/8, into simple, easy-to-follow steps. Whether you're a student tackling homework or just someone who loves a good mathematical puzzle, this guide will help you understand how to solve this equation like a pro. So, grab your pencils, and let's dive in!

Understanding the Equation

Before we jump into solving the equation, let's make sure we understand what it's asking. The equation 31e = 2/8 + 3/8 is an algebraic equation. In this equation, 'e' is a variable, which means it's a symbol (in this case, a letter) that represents an unknown value. Our goal is to find out what value 'e' needs to be in order to make the equation true. On the left side of the equation, we have 31e, which means 31 multiplied by 'e'. On the right side, we have 2/8 + 3/8, which is a sum of two fractions. Understanding these basic components is crucial. Think of it like a puzzle – each piece (number, fraction, variable) has its place, and we need to figure out how they all fit together. The equal sign (=) is the key here; it tells us that whatever is on the left side must be equal to whatever is on the right side. So, our mission is to find the value of 'e' that balances this equation. Now that we have a clear understanding of the equation's structure, let's move on to the first step in solving it: simplifying the fractions.

Simplifying Fractions: The Foundation

Okay, guys, let's kick things off by simplifying the fractions on the right side of our equation: 2/8 + 3/8. This is a crucial first step because it makes the rest of the problem much easier to handle. Remember, fractions represent parts of a whole, and when they have the same denominator (the bottom number), adding them is a breeze. In our case, both fractions have a denominator of 8, which means they're both divided into the same number of parts. When fractions share a common denominator, we can simply add the numerators (the top numbers) while keeping the denominator the same. So, 2/8 + 3/8 becomes (2 + 3)/8. Now, let’s do the math: 2 + 3 equals 5. That means our fraction becomes 5/8. See? We've already made progress! But we’re not stopping there. Always check if the resulting fraction can be simplified further. In this case, 5 and 8 don't share any common factors other than 1, which means 5/8 is in its simplest form. This simplified fraction is much easier to work with in the next steps of our equation. Simplifying fractions is a fundamental skill in algebra, and mastering it will make solving equations like this one so much smoother. So, now that we've got the right side of our equation looking neat and tidy, let's move on to the next step: isolating the variable.

Isolating the Variable: Getting 'e' Alone

Alright, now that we've simplified the fractions, it's time to isolate the variable in our equation, 31e = 5/8. This means we need to get 'e' all by itself on one side of the equation. Think of it like separating a single puzzle piece from the rest of the jumble. In our case, 'e' is currently being multiplied by 31. To undo this multiplication and isolate 'e', we need to perform the opposite operation, which is division. Remember, whatever we do to one side of the equation, we must also do to the other side to keep things balanced. It's like a seesaw – if you add weight to one side, you need to add the same weight to the other side to keep it level. So, to isolate 'e', we'll divide both sides of the equation by 31. This gives us (31e) / 31 = (5/8) / 31. On the left side, the 31s cancel each other out, leaving us with just 'e'. On the right side, we have (5/8) divided by 31. Now, dividing by a whole number is the same as multiplying by its reciprocal (the flipped fraction). So, dividing by 31 is the same as multiplying by 1/31. This transforms our equation into e = (5/8) * (1/31). We’re almost there! Now, let’s move on to the final step: multiplying the fractions and finding the value of 'e'.

Multiplying Fractions: The Final Showdown

Okay, we're in the home stretch now! We've reached the point where we need to multiply the fractions to finally solve for 'e'. Our equation currently looks like this: e = (5/8) * (1/31). Remember, when you multiply fractions, you simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. It's like connecting two pieces of a puzzle – you just fit them together directly. So, let's multiply the numerators: 5 * 1 equals 5. And now let's multiply the denominators: 8 * 31. This might require a little bit of calculation, but don’t worry, we’ve got this! 8 multiplied by 31 is 248. So, our new fraction is 5/248. That means the value of 'e' is 5/248. Now, just like we did with the initial fractions, it's always a good idea to check if this final fraction can be simplified further. In this case, 5 and 248 don't share any common factors other than 1, which means 5/248 is in its simplest form. We’ve done it! We've successfully solved for 'e'. This final step of multiplying fractions is a key skill, and mastering it means you can confidently tackle similar problems in the future. So, let’s recap our journey and see how we arrived at the solution.

Recapping the Solution

Alright, let’s take a step back and recap the entire solution so we can see how all the pieces fit together. We started with the equation 31e = 2/8 + 3/8. The first thing we did was simplify the right side of the equation by adding the fractions 2/8 and 3/8. Since they had the same denominator, we simply added the numerators to get 5/8. This gave us the simplified equation 31e = 5/8. Next, we needed to isolate the variable 'e'. To do this, we divided both sides of the equation by 31, which is the same as multiplying by its reciprocal, 1/31. This gave us e = (5/8) * (1/31). Finally, we multiplied the fractions on the right side by multiplying the numerators (5 * 1 = 5) and the denominators (8 * 31 = 248). This resulted in our final answer: e = 5/248. So, there you have it! We successfully solved the equation. By breaking it down into smaller, manageable steps, we were able to simplify fractions, isolate the variable, and multiply fractions to find the value of 'e'. This step-by-step approach is a powerful tool for tackling any math problem, no matter how daunting it may seem at first. Remember, practice makes perfect, so keep working on these skills and you'll become a math whiz in no time!

Practice Problems

To really nail down these skills, let's try a few practice problems. Solving equations is like riding a bike – the more you practice, the better you get. These problems are similar to the one we just worked through, so you can use the same steps and strategies we discussed. Remember, the key is to break down the problem into smaller parts, simplify fractions, isolate the variable, and perform the necessary operations. Don't be afraid to make mistakes – they're a natural part of the learning process. The important thing is to learn from them and keep pushing forward. So, grab a pencil and paper, and let's put your skills to the test!

1. Solve for x: 25x = 1/4 + 2/4
2. Find the value of y: 16y = 3/10 + 5/10
3. Determine z: 42z = 1/3 + 1/3

Give these a try, and feel free to check your answers with the steps we outlined earlier. If you get stuck, go back and review the sections on simplifying fractions, isolating the variable, and multiplying fractions. Remember, with practice and persistence, you can conquer any math problem that comes your way! Happy solving!