Solving Equations For 6th Grade: Get 20 Points!

Hey guys! Are you ready to dive into the world of equations? Today, we're tackling a problem perfect for 6th graders, and the best part? You can score a cool 20 points! This challenge is designed to help you sharpen your algebra skills, specifically focusing on solving basic equations that involve fractions and a little bit of arithmetic. Let's break down the problem and walk through the steps to find the solution. Don’t worry, it’s not as scary as it sounds! We will explore step-by-step instructions for solving this problem. Let's get started! We're going to tackle the equation: (x - 6/5) + 18/11 = 24/19. Our goal is to find the value of 'x' that makes this equation true. It's all about isolating 'x' on one side of the equation.

First, we need to get rid of the fractions, and there is a standard procedure to solve such equations. We’ll start by working on one side of the equation, aiming to simplify it. In this case, we are going to move the term 18/11 to the right side of the equation. To do this, we subtract 18/11 from both sides. This action will keep the equation balanced. So, our equation now becomes: x - 6/5 = 24/19 - 18/11. Note that the purpose of these operations is to leave the 'x' term alone on one side of the equation. Next, we need to deal with the fractions on the right side. To subtract fractions, they need to have the same denominator. So, we'll find a common denominator for 19 and 11, which is their least common multiple (LCM). The LCM of 19 and 11 is 209 (since 19 and 11 are both prime, their LCM is simply their product). We will convert both fractions to have a denominator of 209. So, 24/19 becomes (24 * 11) / (19 * 11) = 264/209, and 18/11 becomes (18 * 19) / (11 * 19) = 342/209. Now our equation looks like this: x - 6/5 = 264/209 - 342/209. Now, subtracting the numerators gives us x - 6/5 = -78/209. Almost there! Now, to isolate 'x', we need to move -6/5 to the other side. We do this by adding 6/5 to both sides of the equation. Remember, whatever you do to one side, you must do to the other to keep things balanced. So, x = -78/209 + 6/5. Now, we have to solve this new expression of the right side. Just like before, we need a common denominator to add these fractions. The LCM of 209 and 5 is 1045. Convert -78/209 to a fraction with a denominator of 1045: (-78 * 5) / (209 * 5) = -390/1045. Convert 6/5 to a fraction with a denominator of 1045: (6 * 209) / (5 * 209) = 1254/1045. Add the numerators of these fractions: x = -390/1045 + 1254/1045. This gives us x = 864/1045. So, the value of x that solves the equation is 864/1045. This is the solution that would earn you those 20 points! You did it!

Breaking Down the Equation: Understanding the Basics

Alright, let's take a closer look at the core components of this equation and how they interact. Understanding the individual parts will make solving similar problems a breeze. First, we have 'x'. In algebra, 'x' represents an unknown value or a variable. Our mission is to find this unknown. The numbers 6/5, 18/11, and 24/19 are fractions. Fractions represent parts of a whole, and in this context, they are numerical values. The parentheses around (x - 6/5) indicate that we need to perform that operation first, following the order of operations (PEMDAS/BODMAS). The plus and minus signs are the operators. They tell us what mathematical operations to perform – addition and subtraction. Understanding the order of operations is crucial. It's the set of rules that dictate the sequence in which we perform calculations. In our equation, we'll first handle the subtraction and then the addition. Remember, the goal is to isolate 'x' on one side of the equation. This means getting 'x' by itself, with all the other terms moved to the other side. Think of it like a balancing act; whatever you do to one side of the equation, you must do to the other to keep it balanced. This is a fundamental concept in algebra. Now, let's delve into the details of working with fractions. Fractions might seem tricky at first, but they become manageable with practice. Here's the lowdown: To add or subtract fractions, you need to have a common denominator (the same number at the bottom of the fraction). If the fractions don't have a common denominator, you need to find the least common multiple (LCM) of the denominators and convert the fractions accordingly. For example, when we subtract 18/11 from 24/19, we found the LCM of 11 and 19 to be 209. We then converted both fractions to have a denominator of 209. Multiplying the numerator and denominator of a fraction by the same number doesn't change its value. This is how you convert fractions to have a common denominator. Once you have fractions with a common denominator, you can add or subtract their numerators while keeping the denominator the same. When you're dealing with mixed operations, always remember the order of operations. First, deal with parentheses, then exponents (if any), then multiplication and division (from left to right), and finally, addition and subtraction (from left to right). It's essential to stay organized and keep track of each step. Writing down each step will help prevent mistakes and make it easier to follow your logic. Take your time, be methodical, and you'll ace these equations in no time!

Step-by-Step Solution: Your Path to Success

Let's revisit the equation and go through the solution step-by-step, just to make sure everything is crystal clear. This detailed walkthrough will give you a solid grasp of the process, so you can confidently tackle similar problems on your own. The original equation is: (x - 6/5) + 18/11 = 24/19. Step 1: Isolate the 'x' term. Our first move is to get rid of the + 18/11 on the left side of the equation. We do this by subtracting 18/11 from both sides. This keeps the equation balanced. This gives us x - 6/5 = 24/19 - 18/11. Step 2: Subtract the fractions on the right side. To subtract 18/11 from 24/19, we need a common denominator. The LCM of 19 and 11 is 209. Convert the fractions: 24/19 = 264/209 and 18/11 = 342/209. Now we have x - 6/5 = 264/209 - 342/209. Subtracting the numerators gives us x - 6/5 = -78/209. Step 3: Isolate 'x'. Next, we need to move the -6/5 to the other side of the equation. We do this by adding 6/5 to both sides: x = -78/209 + 6/5. Step 4: Add the fractions on the right side. To add -78/209 and 6/5, we need a common denominator. The LCM of 209 and 5 is 1045. Convert the fractions: -78/209 = -390/1045 and 6/5 = 1254/1045. Now we have x = -390/1045 + 1254/1045. Adding the numerators gives us x = 864/1045. Step 5: Simplify if possible. In this case, 864/1045 cannot be simplified further, so our final answer is x = 864/1045. Congratulations! You've solved the equation! Remember, practice makes perfect. The more you work through these types of problems, the easier they will become. Don't hesitate to review these steps and practice with other similar equations. You'll soon find yourself solving them with ease. Each step is a building block, leading you closer to the correct answer. Keep practicing, and you'll be a pro in no time!

Tips and Tricks: Mastering Equation Solving

Alright, let’s equip you with some handy tips and tricks to boost your equation-solving skills. These strategies will make solving equations easier and more enjoyable! First off, practice regularly. The more equations you solve, the more familiar you’ll become with the steps and the more confident you’ll feel. Start with simple equations and gradually increase the difficulty. Next, always double-check your work. Go back through each step to make sure you haven’t made any calculation errors. A simple mistake can lead to an incorrect answer. Checking your work is an essential habit. Write down every step. This helps you stay organized and makes it easier to spot any mistakes. It also allows you to review your work later and understand your thought process. Use the order of operations (PEMDAS/BODMAS) consistently. This will prevent errors and ensure you solve the equation correctly. Don't be afraid to ask for help. If you get stuck, ask your teacher, a classmate, or a parent for assistance. Explaining the problem to someone else can also help you understand it better. Break down complex equations into smaller, more manageable steps. This will make the problem less overwhelming. Focus on one operation at a time, and don't try to do too much in your head. If you're dealing with fractions, make sure to find the correct common denominator before adding or subtracting them. This is a common mistake, so be extra careful with fractions. When working with negative numbers, pay close attention to the signs. A misplaced negative sign can completely change the answer. Get a good grasp of the basics, such as how to add, subtract, multiply, and divide fractions. These skills are fundamental to solving equations. Don't give up! Equation solving can be challenging, but it's a rewarding skill. Persistence is key. Keep practicing, and you'll eventually master it. Celebrate your successes! When you solve an equation correctly, give yourself a pat on the back. This will help you stay motivated and build your confidence. By using these tips and tricks, you’ll be well on your way to becoming an equation-solving expert. Remember, it's all about practice, patience, and a positive attitude. Keep up the great work, and you’ll continue to improve!