Solving Equations: Step-by-Step Guide For (x+3)/2 + (2x)/7 = 7

Hey guys! Let's dive into solving the equation (x+3)/2 + (2x)/7 = 7. This might look intimidating at first, but don't worry! We're going to break it down step by step so it's super easy to understand. We will go over the key concepts you need to master, ensuring you can tackle similar problems confidently. This article provides a comprehensive guide, perfect for anyone looking to improve their algebra skills. Let's get started and make math fun!

Understanding the Basics: Equations and Fractions

Before we jump into the solution, let's quickly refresh some fundamental concepts. An equation is a mathematical statement that asserts the equality of two expressions. Solving an equation means finding the value(s) of the variable(s) that make the equation true. In our case, the variable is 'x'. Dealing with fractions might seem tricky, but they're just numbers! We need to remember how to add fractions and how to clear them from an equation, which we'll see shortly. The core principle in solving any equation is maintaining balance. Whatever operation you perform on one side of the equation, you must perform the same operation on the other side. Think of it like a seesaw – you need to keep it balanced! This ensures the equality remains valid throughout the solution process. We'll apply these principles as we tackle our equation, making each step clear and logical. So, let's get ready to solve this equation with confidence!

Step 1: Clearing the Fractions

Fractions in equations can be a bit messy, so our first goal is to get rid of them. To do this, we'll find the least common multiple (LCM) of the denominators. In our equation, (x+3)/2 + (2x)/7 = 7, the denominators are 2 and 7. The LCM of 2 and 7 is 14 (since 2 and 7 are prime numbers, their LCM is simply their product). Now, we'll multiply both sides of the equation by 14. This is a crucial step because it eliminates the fractions, making the equation much easier to work with. Multiplying each term by 14 gives us: 14 * [(x+3)/2] + 14 * [(2x)/7] = 14 * 7. Simplifying this, we get 7(x+3) + 2(2x) = 98. Notice how the denominators have disappeared! This step highlights the importance of understanding LCM and how it helps simplify algebraic expressions. Clearing fractions is a common technique in solving equations, and mastering it will significantly improve your problem-solving abilities. We've successfully transformed our equation into a cleaner, more manageable form, setting the stage for the next steps in our solution.

Step 2: Distributing and Simplifying

Now that we've cleared the fractions, let's focus on simplifying the equation further. Our equation currently looks like this: 7(x+3) + 2(2x) = 98. The next step involves distributing the numbers outside the parentheses. This means multiplying 7 by both x and 3, and multiplying 2 by 2x. When we distribute, we get: 7x + 21 + 4x = 98. Now, we need to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, 7x and 4x are like terms. Adding them together gives us 11x. So, our equation simplifies to: 11x + 21 = 98. Simplifying expressions is a fundamental skill in algebra, and it's crucial for solving equations efficiently. By distributing and combining like terms, we've reduced the complexity of the equation, bringing us closer to finding the value of x. This step emphasizes the importance of careful algebraic manipulation to ensure accuracy in the solution process. We're making great progress, guys! Let's move on to the next step.

Step 3: Isolating the Variable

Our goal is to get 'x' all by itself on one side of the equation. Currently, we have 11x + 21 = 98. To isolate 'x', we need to get rid of the +21. We can do this by subtracting 21 from both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced! Subtracting 21 from both sides gives us: 11x + 21 - 21 = 98 - 21, which simplifies to 11x = 77. Now, 'x' is almost isolated! We just have the 11 multiplied by it. To get 'x' completely alone, we'll divide both sides of the equation by 11. This gives us: (11x) / 11 = 77 / 11. The 11s on the left side cancel out, leaving us with x = 7. Isolating the variable is a key step in solving equations, and it often involves using inverse operations (like subtraction to undo addition, and division to undo multiplication). We've successfully isolated 'x', and we're just one step away from the final solution! Keep up the great work!

Step 4: Solving for x

We've made it to the final step! After isolating the variable, we have the equation x = 7. This means the solution to our original equation, (x+3)/2 + (2x)/7 = 7, is x = 7. But wait, we're not quite done yet! It's always a good idea to check our answer to make sure it's correct. To do this, we'll substitute x = 7 back into the original equation. So, we have: (7+3)/2 + (2*7)/7 = 7. Simplifying this, we get: 10/2 + 14/7 = 7, which becomes 5 + 2 = 7. And guess what? 7 = 7! This confirms that our solution, x = 7, is correct. Checking our solution is a crucial step because it helps us catch any errors we might have made along the way. By substituting our answer back into the original equation, we can be confident that we've found the correct value for 'x'. Congratulations, guys! We've successfully solved the equation and verified our solution. You're becoming algebra pros!

Conclusion: Mastering Equations

Woohoo! We've successfully solved the equation (x+3)/2 + (2x)/7 = 7, and found that x = 7. We tackled this problem step by step, from clearing fractions to isolating the variable, and finally, verifying our solution. Remember, solving equations is a fundamental skill in mathematics, and the more you practice, the better you'll become. Key takeaways from this exercise include: understanding the importance of the least common multiple (LCM) in clearing fractions, distributing and combining like terms to simplify equations, isolating the variable using inverse operations, and always checking your solution. These steps are applicable to a wide range of algebraic equations, so mastering them will set you up for success in more advanced math courses. Keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this, guys! If you enjoyed this step-by-step guide, be sure to check out more of our math tutorials for additional help and tips. Happy solving!