Solving For X: A Step-by-Step Guide

Nov 6, 2025 by Admin 36 views

$Solving for X: A Step-by-Step Guide to the Equation $\frac{2}{x+5}=\frac{3}{4x-5}$$

Hey math enthusiasts! Today, we're diving into a classic algebra problem: solving for x in the equation $\frac{2}{x+5}=\frac{3}{4x-5}$ . Don't worry if it looks a little intimidating at first. We'll break it down into easy-to-understand steps, making sure everyone can follow along. This is a fundamental concept in algebra, and mastering it will give you a solid foundation for more complex equations. So, grab your pencils and let's get started! We'll cover everything from the basic principles to the final solution, ensuring you grasp the method completely. This guide is designed to be your go-to resource for understanding and solving this type of equation.

Understanding the Basics: Why Solving for X Matters

Before we jump into the equation itself, let's talk about why solving for x is so important. In algebra, x represents an unknown value. Our goal is to find that value. Think of it like a puzzle where x is a missing piece. Once we find x, we can plug it back into the equation to verify that it works. This skill is critical for all kinds of applications, from everyday problem-solving to advanced scientific calculations. Solving for x is used to determine unknown quantities, analyze relationships between variables, and build mathematical models. It's the cornerstone of algebra, forming the basis for more advanced mathematical concepts and real-world applications. When we solve for x, we're not just finding a number; we're unlocking a powerful tool for understanding and manipulating the world around us. Mastering this skill opens doors to countless opportunities in various fields, including science, engineering, economics, and computer science. Therefore, understanding the concepts is very crucial.

Now, let's talk about the equation at hand: $\frac{2}{x+5}=\frac{3}{4x-5}$ . This is a rational equation, meaning it involves fractions where the variable x appears in the denominator. This type of equation requires a specific approach to avoid errors and ensure accuracy in our calculations. Remember that the key is to isolate x on one side of the equation and find its numerical value, which satisfies the original equation. Let's start with a clear understanding of what we're trying to achieve. Our aim is to manipulate the equation legally, following the rules of algebra, until we have an equation of the form x = [some number]. This process will involve several steps, but each one is designed to bring us closer to our goal.

Step-by-Step Solution: Cracking the Code

Alright, let's get to the fun part: solving the equation! Here's a step-by-step guide to help you through the process of solving for x in $\frac{2}{x+5}=\frac{3}{4x-5}$ .

Step 1: Cross-Multiplication. The first step is to get rid of those pesky fractions. We can do this using cross-multiplication. This means multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. So, we get:

2 * (4x - 5) = 3 * (x + 5)

Step 2: Distribute. Next, we need to distribute the numbers outside the parentheses to the terms inside. This simplifies the equation and gets us closer to isolating x.

8x - 10 = 3x + 15

Step 3: Combine Like Terms. Now, we want to get all the x terms on one side of the equation and the constants on the other side. Let's subtract 3x from both sides and add 10 to both sides.

8x - 3x = 15 + 10
5x = 25

Step 4: Isolate x. Finally, to solve for x, we divide both sides of the equation by 5.

x = 25 / 5
x = 5

Step 5: Verify the Solution. The final solution is x = 5. To ensure we have the correct answer, substitute this value back into the original equation to verify.

So, the answer is x = 5. Congratulations, you've solved for x!

This process is the core of solving rational equations. Understanding each step, from cross-multiplication to isolating the variable, is crucial. Remember to double-check your work at each stage to avoid common mistakes. Practicing with similar problems will make you more proficient and confident in tackling algebraic equations.

Common Mistakes and How to Avoid Them

Even the best of us make mistakes! Let's look at some common pitfalls when solving equations like $\frac{2}{x+5}=\frac{3}{4x-5}$ and how to avoid them. Firstly, forgetting to distribute properly is a frequent error. When multiplying a number by a term in parentheses, make sure you multiply every term inside the parentheses. Secondly, mixing up signs is a common mistake. Pay close attention to positive and negative signs, especially when combining like terms. Another common mistake is failing to verify the solution. Always substitute your answer back into the original equation to check if it holds true. This simple step can save you a lot of time and frustration. Finally, being careless with cross-multiplication is a mistake. Make sure you multiply the numerators and denominators correctly.

To avoid these mistakes, always double-check each step of your solution. Write down each step clearly and neatly. This makes it easier to spot errors. Practice regularly to build familiarity and confidence. Check your answer by substituting it back into the original equation. Don't rush; take your time and review your work. By being careful and methodical, you can minimize these errors and improve your problem-solving skills.

Practice Makes Perfect: More Examples and Exercises

Ready to put your skills to the test? Here are a few more examples and exercises to help you practice solving for x in rational equations. Try solving these on your own, and then compare your answers with the solutions provided. This is where you really cement your understanding. Remember, the more you practice, the more confident you'll become! Don't be afraid to make mistakes; that's how we learn. Each problem solved is a step forward in mastering algebra.

$\frac{1}{x-2} = \frac{4}{x+1}$

Solution: x = 3
$\frac{5}{2x+3} = \frac{2}{x-1}$

Solution: x = 11/1
$\frac{3}{x+4} = \frac{6}{2x-1}$

Solution: x = -23/6

Conclusion: Your Journey in Algebra Continues

So there you have it, guys! We have successfully solved for x in the equation $\frac{2}{x+5}=\frac{3}{4x-5}$ . We have also covered some common mistakes and ways to avoid them and have practiced more examples. Remember, algebra is a fundamental skill, and mastering it takes practice and patience. Keep practicing, keep learning, and don't be afraid to ask for help when you need it. Every equation you solve builds your confidence and strengthens your understanding. Your mathematical journey doesn't end here; it continues with every problem you solve and every new concept you explore. Keep up the great work!

I hope this guide has been helpful. Keep up the great work and happy solving!