Solve For X: Unraveling The Equation $\frac{8}{x+9}=\frac{5}{x-3}$

Hey math enthusiasts! Today, we're diving headfirst into the world of algebraic equations. Specifically, we'll be tackling the equation $\frac{8}{x+9}=\frac{5}{x-3}$. Don't worry if it looks a little intimidating at first; we'll break it down step by step, making sure everyone understands how to solve for x. This equation, which involves fractions with variables in the denominators, might seem tricky at first glance. But, trust me, with a few simple algebraic manipulations, we can isolate x and find its value. So, grab your pencils, get comfortable, and let's get started. Our goal is not just to find the answer but to understand the process, so you can confidently solve similar problems in the future. We will carefully navigate through each step, making sure to explain the reasoning behind every action. By the end of this journey, you'll be equipped with the knowledge and the confidence to conquer equations like this. This is more than just finding a number; it is about building your problem-solving skills and reinforcing your grasp of mathematical principles. We will begin by eliminating the fractions, which is often the first step in solving equations of this type. This will involve multiplying both sides of the equation by a common factor. This process simplifies the equation and makes it easier to work with. Following this, we will apply the distributive property to expand the terms and combine like terms. This organized approach to problem-solving ensures accuracy and makes the entire process more understandable. Furthermore, we will then isolate the variable x on one side of the equation. This will involve the application of inverse operations, which essentially “undo” any operations performed on x. These operations could involve adding, subtracting, multiplying, or dividing to isolate x. Finally, we will have a clear solution, where x equals a numerical value. So, are you ready to solve the equation? Let's dive in!

Step-by-Step Solution to Find x

Alright guys, let's get down to the nitty-gritty and solve this equation. The key to solving $\frac{8}{x+9}=\frac{5}{x-3}$ is to get rid of those pesky fractions. Here's how we'll do it. Our first move is to cross-multiply. This means we multiply the numerator of the first fraction by the denominator of the second fraction and vice versa. It is effectively multiplying both sides of the equation by (x+9) and (x-3). So, we have: 8 * (x - 3) = 5 * (x + 9). Now, let's expand the terms using the distributive property. This means we multiply the number outside the parentheses by each term inside the parentheses. So we have, 8 * x - 8 * 3 = 5 * x + 5 * 9. This simplifies to: 8x - 24 = 5x + 45. The next step is to gather all the x terms on one side of the equation and all the constant terms on the other side. Let's subtract 5x from both sides: 8x - 5x - 24 = 5x - 5x + 45. This gives us: 3x - 24 = 45. Now, we add 24 to both sides of the equation to isolate the term with x: 3x - 24 + 24 = 45 + 24. This simplifies to: 3x = 69. Finally, to find the value of x, we divide both sides by 3: 3x / 3 = 69 / 3. So, x = 23. This is our solution! Isn't that great? We've successfully solved for x. Remember, each step here is important, and understanding the reasoning behind each action is crucial for mastering algebraic equations. We began by eliminating the fractions using cross-multiplication, then simplified the equation by using the distributive property. We gathered like terms to isolate x and solve for x. Every step ensures accuracy and makes the solving process more manageable and easy to understand. Keep in mind the importance of the correct application of the distributive property and the correct sign, especially when dealing with negative values. By following these steps consistently, you can accurately solve similar equations.

Detailed Breakdown of Each Step

To make sure we're all on the same page, let's zoom in on each step and clarify the why behind the what. We start with the original equation: $\frac{8}{x+9}=\frac{5}{x-3}$. Cross-multiplication is our first move. By multiplying both sides by (x+9) and (x-3) we effectively get rid of the fractions, making the equation simpler to deal with. This results in 8 * (x - 3) = 5 * (x + 9). The distributive property is our next tool. We multiply the terms outside the parentheses by each term inside. This gives us 8x - 24 = 5x + 45. Always remember to multiply each term correctly, especially with negative numbers. Combining like terms is next. We want all x terms on one side and all constants on the other. Subtracting 5x from both sides gives us 3x - 24 = 45. Then, isolating x becomes our focus. By adding 24 to both sides, we get 3x = 69. Finally, to find the value of x, we divide both sides by 3. This crucial step isolates x, giving us x = 23. Each step is essential. They build upon each other, and the process is easy to master. When you're solving equations, always double-check your calculations, especially the signs. Small mistakes can drastically change the final result. Understanding these steps and the underlying mathematical principles is the key to solving equations like this with confidence. Keep practicing.

Avoiding Common Mistakes

Okay, let's talk about some common pitfalls and how to avoid them. One of the most frequent mistakes is incorrectly applying the distributive property. Always make sure to multiply the term outside the parentheses with every term inside. For instance, in our example, make sure you multiply both 8 by x and 8 by -3. Another common mistake is making sign errors. Keep a close eye on your positive and negative signs, especially when combining like terms and moving them across the equal sign. Remember that when you move a term across the equal sign, you must change its sign. A third common mistake is forgetting to perform operations on both sides of the equation. Whatever operation you do on one side, you must do it on the other to maintain the equation's balance. Skipping this step can lead to an incorrect solution. Always write each step down to avoid simple errors. Checking your work is very helpful. Plug your solution back into the original equation to ensure it's correct. Also, try solving the equation again from scratch. This helps reinforce the steps and identifies any mistakes in your initial approach. Regularly practicing these equations will significantly improve your skills and minimize errors. Remember, practice makes perfect!

Checking Your Work and Next Steps

Alright, we have x = 23. But how do we know if it's the right answer? Checking your solution is crucial. Substitute x = 23 back into the original equation: $\frac{8}{23+9}=\frac{5}{23-3}$. This simplifies to $\frac{8}{32}=\frac{5}{20}$. And, finally, $\frac{1}{4} = \frac{1}{4}$. Since the equation holds true, we know our answer is correct! Now that we have successfully solved the equation, what are the next steps? Now that you've got this one down, why not try solving similar equations? Change the numbers, change the variables, and see how you do. This will help you solidify your understanding and boost your problem-solving confidence. You can also explore more complex equations. Learn to solve equations with different types of fractions or those that involve exponents, radicals, or other advanced concepts. This expansion can greatly enhance your mathematical capabilities. Also, practice regularly! Solve different types of equations every day. This will reinforce what you've learned. In addition to this, remember to always revisit the fundamental concepts. When you encounter difficulties, go back to the basic principles of algebra. This will help you identify the area where your understanding needs improvement. Lastly, don't be afraid to ask for help! Whenever you have trouble, don't hesitate to seek advice from teachers, classmates, or online resources. Learning is much easier when you have help. Good luck, and keep practicing!

Conclusion: Mastering Algebraic Equations

We did it, guys! We have successfully navigated the equation $\frac{8}{x+9}=\frac{5}{x-3}$! I hope this step-by-step guide has demystified the process and equipped you with the confidence to tackle similar equations. Remember the key steps: cross-multiplication, distributing, combining like terms, isolating the variable, and always checking your work. Keep practicing, don't be afraid to make mistakes (it's part of the learning process!), and most importantly, enjoy the journey of learning. Math is like a puzzle, and solving it is one of the most fulfilling feelings. By understanding the underlying principles and continuously practicing, you'll become more confident and capable of solving complex problems. I hope this was helpful. Keep practicing and exploring the fascinating world of mathematics! You've got this!