Solving 4x/5 - 3/10 = X/5 + X/4 A Step-by-Step Guide

In this comprehensive guide, we will delve into the step-by-step process of solving for 'X' in the equation 4x/5 - 3/10 = X/5 + X/4. This equation combines fractions, variables, and basic arithmetic operations, making it a great example for understanding algebraic problem-solving. Our approach will be methodical, ensuring clarity and ease of understanding for anyone looking to master such equations. This detailed explanation is designed not only to provide the solution but also to enhance your understanding of the underlying mathematical principles. We will break down each step, explaining the logic and the mathematical rules applied. Whether you're a student tackling algebra problems or someone looking to refresh their math skills, this guide will provide valuable insights and practical steps for solving similar equations. By the end of this article, you will have a clear understanding of how to manipulate algebraic equations involving fractions, and you will be equipped with the knowledge to solve similar problems with confidence. Let's embark on this mathematical journey together and unlock the solution to this equation, paving the way for tackling more complex algebraic challenges in the future.

1. Understanding the Equation: 4x/5 - 3/10 = X/5 + X/4

Before diving into the solution, it's crucial to understand the structure of the equation. We have an equation that involves fractions, a variable 'X', and basic arithmetic operations such as subtraction and addition. The equation 4x/5 - 3/10 = X/5 + X/4 presents a balanced relationship between the left-hand side (LHS) and the right-hand side (RHS). Our goal is to isolate 'X' on one side of the equation to determine its value. To achieve this, we will employ algebraic techniques that involve manipulating the equation while maintaining its balance. This involves performing the same operations on both sides of the equation. Understanding the properties of equality is paramount in solving algebraic equations. This principle states that if we perform the same operation (addition, subtraction, multiplication, or division) on both sides of an equation, the equality remains true. This allows us to rearrange terms, combine like terms, and eventually isolate the variable we are solving for. In this specific equation, we will be focusing on clearing the fractions to simplify the equation and make it easier to work with. This will involve finding a common denominator and multiplying each term by it. This step is crucial in simplifying the equation and bringing us closer to the solution. Let's break down each part of the equation and see how they interact with each other to form this mathematical puzzle. By understanding the components, we can then strategically apply algebraic principles to solve for 'X'.

2. Clearing the Fractions: Finding the Least Common Multiple (LCM)

The presence of fractions can make an equation appear more complex, but we can simplify it by clearing these fractions. The key to doing this is to find the Least Common Multiple (LCM) of the denominators. In our equation, 4x/5 - 3/10 = X/5 + X/4, the denominators are 5, 10, and 4. The LCM is the smallest number that is a multiple of all these denominators. To find the LCM of 5, 10, and 4, we can list the multiples of each number: Multiples of 5: 5, 10, 15, 20, 25, ... Multiples of 10: 10, 20, 30, 40, ... Multiples of 4: 4, 8, 12, 16, 20, 24, ... From the lists, we can see that the smallest multiple common to all three numbers is 20. Therefore, the LCM of 5, 10, and 4 is 20. Now that we have the LCM, we can multiply both sides of the equation by 20. This will clear the fractions, making the equation easier to manipulate. Multiplying each term by the LCM is a fundamental step in solving equations with fractions. This step ensures that we eliminate the denominators and work with whole numbers, which simplifies the algebraic manipulations that follow. By carefully calculating the LCM and applying it to the equation, we set the stage for solving for 'X' in a more straightforward manner. This technique is a cornerstone of algebra and is applicable to a wide range of equations involving fractions.

3. Multiplying the Equation by the LCM: Eliminating Denominators

Now that we have determined the LCM to be 20, the next step is to multiply each term in the equation 4x/5 - 3/10 = X/5 + X/4 by 20. This process will eliminate the denominators, simplifying the equation. Let's perform the multiplication: 20 * (4x/5) - 20 * (3/10) = 20 * (X/5) + 20 * (X/4) Now, we simplify each term: (20/5) * 4x - (20/10) * 3 = (20/5) * X + (20/4) * X 4 * 4x - 2 * 3 = 4 * X + 5 * X This simplifies further to: 16x - 6 = 4x + 5x This equation is now free of fractions, making it much easier to work with. The process of multiplying by the LCM is crucial because it transforms the equation into a more manageable form. By eliminating the denominators, we can focus on the coefficients and variables, which allows us to isolate 'X' more effectively. This step is not just about simplifying the equation; it's about making the algebraic manipulations clearer and reducing the chances of errors. The resulting equation, 16x - 6 = 4x + 5x, is a linear equation that we can solve using basic algebraic techniques. This transformation is a testament to the power of using the LCM to clear fractions in equations, a technique that is widely used in algebra and beyond.

4. Simplifying the Equation: Combining Like Terms

After clearing the fractions, our equation is now 16x - 6 = 4x + 5x. The next step is to simplify the equation by combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have '4x' and '5x' on the right-hand side of the equation, which are like terms. Let's combine these terms: 4x + 5x = 9x So, the equation becomes: 16x - 6 = 9x Now, our goal is to isolate 'X' on one side of the equation. To do this, we need to move the '9x' term from the right-hand side to the left-hand side. We can do this by subtracting '9x' from both sides of the equation: 16x - 6 - 9x = 9x - 9x This simplifies to: 7x - 6 = 0 Combining like terms is a fundamental step in solving algebraic equations. It allows us to reduce the number of terms in the equation, making it simpler to manipulate and solve. This step is crucial for clarity and accuracy in the solution process. By combining like terms, we streamline the equation and bring it closer to a form where we can easily isolate the variable. The simplified equation, 7x - 6 = 0, is a linear equation that is now much easier to solve. This process of simplification is a key skill in algebra, enabling us to tackle more complex equations by breaking them down into manageable parts.

5. Isolating the Variable: Moving Constant Terms

Our equation is currently in the form 7x - 6 = 0. To isolate the variable 'X', we need to move the constant term, which is '-6', to the other side of the equation. We can do this by adding 6 to both sides of the equation: 7x - 6 + 6 = 0 + 6 This simplifies to: 7x = 6 Now, we have the term with 'X' isolated on the left-hand side, and a constant term on the right-hand side. The next step is to solve for 'X' by dividing both sides of the equation by the coefficient of 'X', which is 7. Isolating the variable is a crucial step in solving algebraic equations. It involves strategically manipulating the equation to get the variable term alone on one side. This often involves performing inverse operations, such as adding or subtracting constants, to move terms across the equals sign. By isolating 'X', we bring the equation to a form where we can directly solve for its value. The equation 7x = 6 represents a significant milestone in our solution process. We have successfully separated the variable term from the constants, paving the way for the final step of dividing to find the value of 'X'. This step demonstrates the power of algebraic manipulation in isolating variables and simplifying equations.

6. Solving for X: Dividing to Find the Solution

We have arrived at the equation 7x = 6. To solve for 'X', we need to divide both sides of the equation by the coefficient of 'X', which is 7: (7x) / 7 = 6 / 7 This simplifies to: X = 6/7 Therefore, the solution to the equation 4x/5 - 3/10 = X/5 + X/4 is X = 6/7. This is the value of 'X' that makes the equation true. The process of dividing to find the solution is the final step in solving for a variable in a linear equation. It involves undoing the multiplication that is occurring between the variable and its coefficient. By dividing both sides of the equation by the coefficient, we isolate the variable and determine its value. The solution, X = 6/7, represents the answer to our original equation. This value, when substituted back into the original equation, will satisfy the equality. This final step underscores the importance of understanding the inverse operations in algebra and how they are used to solve for unknowns. The solution X = 6/7 is a fraction, which is a common outcome in algebra. It's important to express the solution in its simplest form, and in this case, 6/7 is already in its simplest form. We have successfully navigated the equation, step by step, and arrived at the solution, demonstrating the power of algebraic techniques in problem-solving.

7. Verification: Substituting the Value of X Back into the Original Equation

To ensure that our solution is correct, it's always a good practice to verify it. We do this by substituting the value of X back into the original equation, 4x/5 - 3/10 = X/5 + X/4, and checking if both sides of the equation are equal. Our solution is X = 6/7. Let's substitute this value into the equation: 4 * (6/7) / 5 - 3/10 = (6/7) / 5 + (6/7) / 4 Now, let's simplify each term: (24/7) / 5 - 3/10 = 6/35 + 6/28 This simplifies to: 24/35 - 3/10 = 6/35 + 3/14 To compare the two sides, we need to find a common denominator for each side. On the left-hand side, the common denominator for 35 and 10 is 70. On the right-hand side, the common denominator for 35 and 14 is 70. Converting the fractions, we get: (24/35) * (2/2) - (3/10) * (7/7) = (6/35) * (2/2) + (3/14) * (5/5) 48/70 - 21/70 = 12/70 + 15/70 This simplifies to: 27/70 = 27/70 Since both sides of the equation are equal, our solution X = 6/7 is correct. Verification is a critical step in the problem-solving process. It ensures that the solution we have found is accurate and satisfies the original equation. By substituting the value of X back into the equation, we can confirm that both sides of the equation balance out. This step is not just about checking our work; it's about building confidence in our solution and understanding the underlying principles of algebra. The verification process also reinforces the concept of equality in equations, where both sides must remain balanced when performing operations. In this case, the successful verification validates our step-by-step solution and confirms the correctness of X = 6/7. This thorough approach to problem-solving, including verification, is a valuable skill in mathematics and beyond.

Conclusion

In conclusion, we have successfully solved the equation 4x/5 - 3/10 = X/5 + X/4 by systematically applying algebraic principles. Our journey began with understanding the equation, followed by clearing the fractions using the Least Common Multiple (LCM). We then multiplied the equation by the LCM, simplified by combining like terms, and isolated the variable 'X'. Finally, we solved for 'X' and verified our solution by substituting it back into the original equation. The solution we found is X = 6/7. This process demonstrates the importance of a step-by-step approach in solving algebraic equations. Each step builds upon the previous one, leading us closer to the solution. The techniques we used, such as clearing fractions, combining like terms, and isolating the variable, are fundamental in algebra and are applicable to a wide range of problems. The verification step underscores the importance of accuracy and attention to detail in mathematics. By checking our solution, we ensure that it satisfies the original equation and that our work is correct. This comprehensive guide not only provides the solution to the equation but also explains the underlying concepts and techniques. This understanding is crucial for tackling more complex algebraic problems in the future. Solving equations like this enhances our problem-solving skills and builds a strong foundation in mathematics. The ability to manipulate equations, isolate variables, and solve for unknowns is a valuable skill that extends beyond the classroom and into various real-world applications. This journey through the equation serves as a testament to the power of algebraic thinking and its role in solving mathematical challenges.