Solving X/4 – 7/5 + X/10 = X/2 + 7/20 A Step-by-Step Guide

by Viktoria Ivanova 59 views

Hey guys! Today, we're going to tackle a common algebraic equation: X/4 – 7/5 + X/10 = X/2 + 7/20. Don't worry, it might look a little intimidating at first, but we'll break it down step by step so everyone can follow along. Whether you're a student brushing up on your algebra skills or just curious about how to solve these equations, this guide is for you. We'll go through each stage meticulously, ensuring you grasp not just the 'how' but also the 'why' behind each operation. So, grab your pencils and paper, and let’s dive into the world of algebra! This detailed walkthrough is designed to help you master the art of solving linear equations, which forms the bedrock of more advanced mathematical concepts. Understanding the underlying principles will empower you to approach similar problems with confidence and accuracy. We'll not only solve this specific equation but also highlight general strategies and techniques that are applicable to a wide range of algebraic challenges. Remember, math isn't just about finding the correct answer; it’s about understanding the process and developing a logical approach to problem-solving. As we progress, we'll emphasize the importance of maintaining balance and order in equations, which are crucial for avoiding errors and achieving the correct solution. Each step is carefully explained to promote clarity and comprehension. So, let's embark on this algebraic adventure together and unlock the mystery behind solving equations! With a bit of patience and practice, you'll find that even the most complex equations can be tamed.

1. Understanding the Equation

Before we jump into solving, let’s make sure we understand what the equation X/4 – 7/5 + X/10 = X/2 + 7/20 actually means. In simple terms, we're looking for the value of 'X' that makes both sides of the equation equal. Think of it like a balance scale; our goal is to keep both sides perfectly balanced. Each term in the equation, whether it's X/4, -7/5, X/10, X/2, or 7/20, represents a specific quantity. The equal sign (=) is the pivotal point, signifying that the sum of the quantities on the left must be identical to the sum of the quantities on the right. To find 'X', we need to isolate it on one side of the equation. This involves performing a series of operations that maintain the balance. These operations typically include addition, subtraction, multiplication, and division. The key is to apply the same operation to both sides of the equation to ensure the balance remains intact. Understanding this fundamental principle is crucial for solving any algebraic equation. It's like having a blueprint before starting a construction project; a clear understanding of the equation’s structure helps us strategize our approach. We'll break down the equation into its components, analyze each term, and then develop a plan to simplify and solve for 'X'. Remember, patience and attention to detail are your best allies in this process. By understanding the equation thoroughly, we set ourselves up for success in the subsequent steps.

2. Finding the Least Common Denominator (LCD)

The first step in solving this equation is to deal with those pesky fractions. To do this effectively, we need to find the Least Common Denominator (LCD) of all the fractions in the equation: X/4, 7/5, X/10, X/2, and 7/20. The LCD is the smallest number that all the denominators (4, 5, 10, 2, and 20) can divide into evenly. So, how do we find it? We list the multiples of each denominator: Multiples of 4: 4, 8, 12, 16, 20, 24,... Multiples of 5: 5, 10, 15, 20, 25,... Multiples of 10: 10, 20, 30,... Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20,... Multiples of 20: 20, 40, 60,... Looking at these lists, we can see that the smallest number that appears in all of them is 20. Therefore, the LCD is 20. Why is finding the LCD so important? Because it allows us to eliminate the fractions by multiplying every term in the equation by this common denominator. This simplifies the equation and makes it much easier to solve. It's like converting different units of measurement to a common unit before adding or subtracting them. Once we have the LCD, we can proceed to the next step, which involves multiplying each term by 20. This process effectively clears the fractions, transforming the equation into a more manageable form. Understanding and finding the LCD is a crucial skill in algebra, and it's a technique that will serve you well in many mathematical contexts.

3. Multiplying Both Sides by the LCD

Now that we've found the LCD, which is 20, we're going to multiply every single term in the equation X/4 – 7/5 + X/10 = X/2 + 7/20 by 20. This is a crucial step because it eliminates the fractions, making the equation much simpler to work with. Remember, whatever we do to one side of the equation, we must do to the other side to keep the balance intact. Let's go through each term: 20 * (X/4) = 5X (because 20 divided by 4 is 5) 20 * (-7/5) = -28 (because 20 divided by 5 is 4, and 4 times -7 is -28) 20 * (X/10) = 2X (because 20 divided by 10 is 2) 20 * (X/2) = 10X (because 20 divided by 2 is 10) 20 * (7/20) = 7 (because 20 divided by 20 is 1) So, after multiplying each term by 20, our equation now looks like this: 5X - 28 + 2X = 10X + 7. See how much cleaner that looks? No more fractions! This step is like clearing away the clutter before starting a project; it makes the task at hand much more manageable. By multiplying by the LCD, we've transformed a complex equation with fractions into a simple linear equation. This is a common technique used in algebra, and mastering it will significantly improve your problem-solving abilities. The next step involves combining like terms, which will further simplify the equation and bring us closer to isolating 'X'.

4. Combining Like Terms

Alright, we've gotten rid of the fractions, and our equation now looks like 5X - 28 + 2X = 10X + 7. The next step is to combine like terms on each side of the equation. Like terms are terms that have the same variable raised to the same power. In this case, we have 'X' terms and constant terms. On the left side of the equation, we have 5X and 2X, which are like terms. Combining them gives us 5X + 2X = 7X. We also have the constant term -28. So, the left side of the equation simplifies to 7X - 28. On the right side of the equation, we have 10X, which is an 'X' term, and 7, which is a constant term. There are no other like terms to combine on this side, so it remains 10X + 7. Now, our equation looks even simpler: 7X - 28 = 10X + 7. We've reduced the number of terms in the equation, making it easier to isolate 'X'. Combining like terms is like organizing your tools before starting a job; it streamlines the process and prevents confusion. By grouping similar terms together, we've created a clearer picture of the equation and made it easier to see the next steps. This is a fundamental algebraic technique that's essential for solving equations efficiently. The next step involves moving all the 'X' terms to one side of the equation and the constant terms to the other side. This will bring us closer to isolating 'X' and finding its value.

5. Isolating the Variable

Now that our equation is simplified to 7X - 28 = 10X + 7, it's time to isolate the variable 'X'. This means we want to get all the 'X' terms on one side of the equation and all the constant terms on the other side. Let's start by moving the 'X' terms. We can subtract 7X from both sides of the equation: 7X - 28 - 7X = 10X + 7 - 7X This simplifies to: -28 = 3X + 7 Now, let's move the constant terms. We can subtract 7 from both sides of the equation: -28 - 7 = 3X + 7 - 7 This simplifies to: -35 = 3X We're almost there! Now we have 'X' terms on one side and constant terms on the other. To isolate 'X' completely, we need to get rid of the coefficient 3 that's multiplying 'X'. We can do this by dividing both sides of the equation by 3: -35 / 3 = 3X / 3 This simplifies to: X = -35/3 So, we've successfully isolated 'X'! The value of 'X' that satisfies the equation is -35/3. Isolating the variable is like solving a puzzle; each step brings us closer to the final answer. By carefully moving terms and performing operations on both sides of the equation, we've managed to get 'X' by itself. This is a crucial skill in algebra and is the key to solving a wide range of equations. In the next step, we'll check our solution to make sure it's correct.

6. Checking the Solution

We've found that X = -35/3, but it's always a good idea to check our solution to make sure it's correct. To do this, we'll substitute -35/3 back into the original equation: X/4 – 7/5 + X/10 = X/2 + 7/20 Substituting X = -35/3: (-35/3)/4 – 7/5 + (-35/3)/10 = (-35/3)/2 + 7/20 Let's simplify each term: (-35/3)/4 = -35/12 -7/5 remains -7/5 (-35/3)/10 = -35/30 = -7/6 (-35/3)/2 = -35/6 7/20 remains 7/20 Now our equation looks like: -35/12 – 7/5 - 7/6 = -35/6 + 7/20 To check if both sides are equal, we need to find a common denominator for all the fractions, which is 60. Converting each fraction to have a denominator of 60: (-35/12) * (5/5) = -175/60 (-7/5) * (12/12) = -84/60 (-7/6) * (10/10) = -70/60 (-35/6) * (10/10) = -350/60 (7/20) * (3/3) = 21/60 Now our equation looks like: -175/60 - 84/60 - 70/60 = -350/60 + 21/60 Let's add the fractions on each side: Left side: (-175 - 84 - 70)/60 = -329/60 Right side: (-350 + 21)/60 = -329/60 Both sides are equal! This confirms that our solution X = -35/3 is correct. Checking our solution is like proofreading a document; it ensures that we haven't made any mistakes along the way. It's a crucial step in the problem-solving process and gives us confidence in our answer. By substituting our solution back into the original equation and verifying that both sides are equal, we can be sure that we've solved the equation correctly.

Conclusion

Great job, guys! We've successfully solved the equation X/4 – 7/5 + X/10 = X/2 + 7/20 step by step. We found the LCD, multiplied both sides by the LCD to eliminate fractions, combined like terms, isolated the variable, and finally, checked our solution. The value of X that satisfies the equation is -35/3. Solving algebraic equations can be challenging, but by breaking them down into smaller, manageable steps, we can tackle even the most complex problems. Remember, practice makes perfect! The more equations you solve, the more comfortable you'll become with the process. Each step we took – from finding the LCD to checking our solution – is a valuable tool in your algebraic arsenal. These techniques are not just limited to this specific equation; they can be applied to a wide range of algebraic problems. Understanding the underlying principles and practicing consistently will empower you to approach any equation with confidence. Algebra is a fundamental part of mathematics, and mastering it opens doors to more advanced concepts and applications. So, keep practicing, keep exploring, and never be afraid to tackle a new challenge. You've got this!