Solve: -(4x-1)/5 + (x+4)/2 = 3 | Math Guide
Introduction
Linear equations are fundamental in mathematics, and mastering the techniques to solve them is crucial for more advanced topics. Guys, in this article, we will dive deep into solving the linear equation $ -\frac{4x-1}{5} + \frac{x+4}{2} = 3 $. We'll break down each step with detailed explanations to ensure you understand not just the how but also the why behind each operation. So, buckle up and let’s get started! This article is designed to be your ultimate guide, providing a comprehensive understanding of how to tackle similar problems in the future. By the end of this guide, you'll be equipped with the skills and confidence to solve linear equations like a pro. We’ll cover everything from the initial setup to the final solution, including common pitfalls and how to avoid them. Whether you're a student prepping for an exam or just brushing up on your math skills, this article has something for everyone. Remember, practice makes perfect, so feel free to work through the examples as we go along. Let’s make math fun and accessible together! We’ll also touch on the importance of checking your solutions to ensure accuracy, a critical step in problem-solving. So, let’s embark on this mathematical journey together and conquer this linear equation!
Understanding the Equation
Before we jump into solving, let’s first understand the equation we’re dealing with: $ -\frac{4x-1}{5} + \frac{x+4}{2} = 3 $. This equation is a linear equation because the highest power of the variable x is 1. To solve it, our goal is to isolate x on one side of the equation. This involves several algebraic manipulations, such as eliminating fractions, combining like terms, and using inverse operations. Don't worry if this sounds complicated now; we'll take it one step at a time. The key to success with linear equations is to follow the order of operations and keep the equation balanced. Whatever you do to one side, you must do to the other. Think of it like a seesaw; you need to maintain equilibrium. Each term in the equation plays a crucial role, and understanding these roles will make the solving process much smoother. We have fractions, terms with x, and constants, all working together to form this equation. By carefully dissecting each part, we can develop a strategy to solve for x. So, let’s get ready to roll up our sleeves and dive into the nitty-gritty details of this equation!
Step 1: Eliminating Fractions
Fractions can make equations look intimidating, but there’s a simple trick to get rid of them: multiply both sides of the equation by the least common multiple (LCM) of the denominators. In our case, the denominators are 5 and 2. The LCM of 5 and 2 is 10. So, we multiply both sides of the equation by 10:
$ 10 \times \left(-\frac{4x-1}{5} + \frac{x+4}{2}\right) = 10 \times 3 $.
This simplifies to:
$ -2(4x-1) + 5(x+4) = 30 $.
Notice how multiplying by the LCM clears the fractions, making the equation much easier to work with. This is a crucial step in solving equations with fractions. By eliminating the fractions, we transform the equation into a more manageable form, allowing us to focus on the algebraic manipulations needed to isolate x. Think of it as paving the road for the rest of the solution. This technique is widely used in algebra, and mastering it will greatly enhance your problem-solving skills. So, remember, when you see fractions in an equation, your first instinct should be to clear them by multiplying by the LCM. This sets the stage for a smoother, more straightforward solution process. Now, let’s move on to the next step!
Step 2: Expanding and Simplifying
Now that we've eliminated the fractions, let’s expand the terms and simplify the equation. We have: $ -2(4x-1) + 5(x+4) = 30 $. Distribute the -2 and the 5:
$ -8x + 2 + 5x + 20 = 30 $.
Now, combine like terms (terms with x and constant terms):
$ (-8x + 5x) + (2 + 20) = 30 $.
This simplifies to:
$ -3x + 22 = 30 $.
Expanding and simplifying is a key step in solving equations. It helps to organize the equation and make it easier to isolate the variable. By distributing the constants and combining like terms, we reduce the complexity of the equation, bringing us closer to the solution. This process is like tidying up a messy room; once everything is organized, it’s much easier to find what you’re looking for. Each term now has its place, and we can see clearly the next steps to take. Remember, accuracy is crucial in this step. A small mistake in expanding or combining terms can lead to a wrong answer. So, take your time, double-check your work, and ensure every step is correct. With the equation simplified, we’re now ready to isolate x and find our solution!
Step 3: Isolating the Variable
Our goal now is to isolate x. We have the equation: $ -3x + 22 = 30 $. To isolate x, we first subtract 22 from both sides:
$ -3x + 22 - 22 = 30 - 22 $.
This gives us:
$ -3x = 8 $.
Next, we divide both sides by -3:
$ \frac{-3x}{-3} = \frac{8}{-3} $.
This results in:
$ x = -\frac{8}{3} $.
Isolating the variable is the heart of solving any equation. It's the process of getting x by itself on one side, allowing us to see its value. This step often involves inverse operations, which are operations that undo each other. In our case, we used subtraction to undo addition and division to undo multiplication. Think of it as peeling back the layers of an onion to get to the core. Each operation we perform brings us closer to revealing the value of x. It’s crucial to maintain balance in this step, ensuring that whatever operation you perform on one side, you also perform on the other. This keeps the equation true and preserves the solution. So, with x isolated, we’ve found our solution! But before we celebrate, let’s move on to the final step: checking our answer.
Step 4: Checking the Solution
It’s always a good idea to check your solution to make sure it’s correct. We found that $ x = -\frac8}{3} $. Let’s substitute this value back into the original equation{5} + \frac{x+4}{2} = 3 $.
Substituting $ x = -\frac{8}{3} $, we get:
$ -\frac{4(-\frac{8}{3})-1}{5} + \frac{-\frac{8}{3}+4}{2} $.
Simplify the expression:
$ -\frac{-\frac{32}{3}-1}{5} + \frac{-\frac{8}{3}+\frac{12}{3}}{2} $.
$ -\frac{-\frac{35}{3}}{5} + \frac{\frac{4}{3}}{2} $.
$ \frac{7}{3} + \frac{2}{3} $.
$ \frac{9}{3} = 3 $.
Since the left side equals the right side, our solution is correct. Checking your solution is like proofreading a paper; it helps you catch any mistakes and ensures your answer is accurate. It’s a crucial step in the problem-solving process and gives you confidence in your result. By substituting the solution back into the original equation, we verify that it satisfies the equation. This not only confirms our answer but also reinforces our understanding of the problem-solving process. So, always make time to check your solution; it’s a small investment that pays big dividends in accuracy and confidence!
Conclusion
Guys, we’ve successfully solved the linear equation $ -\frac{4x-1}{5} + \frac{x+4}{2} = 3 $. We found that $ x = -\frac{8}{3} $. Remember, the key steps are eliminating fractions, expanding and simplifying, isolating the variable, and checking the solution. Linear equations are a cornerstone of algebra, and mastering them opens doors to more advanced mathematical concepts. This journey through solving this equation has provided us with valuable insights and skills that will serve us well in future mathematical endeavors. From understanding the importance of LCM to the necessity of checking our solutions, each step has reinforced best practices in problem-solving. So, keep practicing, keep exploring, and keep pushing your mathematical boundaries. With each equation you solve, you’re not just finding an answer; you’re building a foundation of knowledge and skills that will empower you to tackle any mathematical challenge that comes your way. Keep up the great work, and remember, math can be fun!
This detailed walkthrough should give you a solid understanding of how to solve similar linear equations. Keep practicing, and you’ll become a pro in no time!
