Solving Equations: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the exciting world of solving equations. Specifically, we'll tackle two problems, breaking down each step to make the process super clear. Let's get started, shall we? This guide is designed to help you, whether you're a student struggling with homework or just a curious mind. We'll explore two equations: (a) x:2.4=12.4:6.2 and (b) 2 5/6:x=1 1/6: 1/17. Our goal is to find the value of 'x' in each equation. Remember, solving equations is like uncovering a hidden treasure – with the right tools and a bit of patience, you'll find the answer! Let's get cracking!

Equation (a): x:2.4=12.4:6.2

Alright, guys, let's start with equation (a). We have x:2.4=12.4:6.2. This equation is essentially a proportion, which means that the ratio on the left side of the equal sign is the same as the ratio on the right side. Our mission? To isolate 'x' and find its value. The key here is to understand the concept of division and how to manipulate the equation to get 'x' by itself. We are going to isolate x to get the final answer. To isolate x we must multiply both sides of the equation by 2.4. So, let's break it down step-by-step:

First, we rewrite the equation as x/2.4 = 12.4/6.2. This is the same equation, just written in a more familiar fraction form. Next, we can simplify the right side of the equation. Divide 12.4 by 6.2, and you get 2. So, our equation now looks like this: x/2.4 = 2. Now, to get 'x' by itself, we need to get rid of the division by 2.4. To do that, we multiply both sides of the equation by 2.4. This cancels out the 2.4 on the left side, leaving us with x = 2 * 2.4. And finally, multiply 2 by 2.4 to get your answer: x = 4.8. So, the solution to equation (a) is x = 4.8. Boom! You've solved your first equation. It's like a puzzle, and you've found the missing piece. Keep up the great work!

This process, though simple, is fundamental to algebra. The ability to manipulate equations by performing the same operations on both sides is crucial. This ensures that the equation remains balanced, allowing us to accurately solve for the unknown variable. Understanding proportions is also essential. Proportions are used everywhere, from scaling recipes to understanding maps. They represent an equivalence between two ratios, which is the foundation of many mathematical concepts. Remember, practice makes perfect. The more equations you solve, the more comfortable and confident you'll become. Each equation you solve is a victory. It's proof of your growing mathematical skills and problem-solving abilities. Don't be afraid to make mistakes; they are a part of the learning process. Learning math is a journey, and every step counts. Celebrate your progress and enjoy the ride!

Equation (b): 2 5/6:x=1 1/6: 1/17

Now, let's move on to equation (b): 2 5/6:x=1 1/6: 1/17. This equation involves fractions, so we'll need to brush up on our fraction skills. No worries, though, it's not as scary as it sounds! The core strategy remains the same: isolate 'x' to find its value. But before we begin, let's convert the mixed numbers into improper fractions. The mixed number 2 5/6 becomes (2 * 6 + 5)/6 = 17/6. And, the mixed number 1 1/6 becomes (1 * 6 + 1)/6 = 7/6. Let's rewrite our equation with these improper fractions:

Our equation now looks like this: (17/6) / x = (7/6) / (1/17). This step simplifies things, making the equation easier to handle. Next, we must convert division by a fraction into multiplication by its reciprocal. The reciprocal of a fraction is simply flipping the numerator and denominator. When we say (17/6) / x, this can be written as (17/6) * (1/x). And, (7/6) / (1/17) becomes (7/6) * (17/1). So our equation now looks like this: (17/6) * (1/x) = (7/6) * 17. Now we need to solve for x, so we will multiply both sides by x and divide both sides by (7/6) * 17. The equation now looks like this (17/6) = (7/6) * 17 * x, and now we will isolate x to get x = (17/6) / ((7/6) * 17). Simplify the right side by multiplying (7/6) by 17, and we get 119/6. So the equation is now x = (17/6) / (119/6). Now divide the fractions, and we get x = (17/6) * (6/119). Multiply and reduce the fraction. So, 176 = 102 and 6119 = 714, and the equation now looks like x = 102/714. Now we reduce it, and we finally get x = 1/7. There you have it! The solution to equation (b) is x = 1/7. Congrats on tackling this equation with fractions! You should feel proud of what you've accomplished. You did a great job!

Dealing with fractions can seem complicated at first, but with practice, it becomes second nature. Converting mixed numbers to improper fractions simplifies calculations. Also, remember the rule of division: dividing by a fraction is the same as multiplying by its reciprocal. This is a super handy trick. It's all about keeping track of the steps. Don’t get discouraged if you need to review the rules for fractions. It's essential to understand these basic concepts to move forward with more complex problems. Always double-check your work, and don't hesitate to ask for help if you need it. Math is a building process. Each new concept builds upon the previous ones. Keep practicing, and your confidence will soar. Remember that every step you take brings you closer to mastering math. You're developing critical thinking skills that will benefit you in all areas of life. Be patient with yourself, celebrate your successes, and keep learning!

Conclusion: Mastering Equation Solving

Well, guys, we've successfully solved both equations! You've learned how to isolate the variable 'x' in both simple and more complex equations. We've reviewed fundamental concepts like proportions and fractions. This is a big win! You're now equipped with the basic skills needed to tackle a wide variety of algebraic problems. Remember, the key to success in math is practice. Continue working through different types of problems to solidify your understanding. Experiment with various examples and don't be afraid to ask questions. There are tons of resources available, including online tutorials, textbooks, and practice quizzes.

Always remember to double-check your work. This helps you catch any mistakes you might have made along the way. Be sure to show your work step-by-step so that you can go back and check where you might have made an error. Look for patterns and shortcuts. The more you work with equations, the more familiar you'll become with different strategies and techniques. Enjoy the process of learning and discovery. Each problem solved is a testament to your growing skills and determination. Keep up the fantastic work, and keep exploring the amazing world of mathematics! You've got this! Math can be super fun, and the more you practice the better you get. Believe in yourself and celebrate every step of the way!