Solving For X In The Equation 2.5(6x - 4) = 10 + 4(1.5 + 0.5x)

Introduction

In this article, we will delve into solving a linear equation for the variable x. Linear equations are fundamental in mathematics and have wide applications in various fields, including physics, engineering, economics, and computer science. Understanding how to solve these equations is a crucial skill for anyone studying or working in these areas. The specific equation we will address is: 2.5(6x - 4) = 10 + 4(1.5 + 0.5x). This equation may seem complex at first glance, but by systematically applying the principles of algebra, we can simplify it and isolate x to find its value. The process involves distributing, combining like terms, and using inverse operations to maintain the equation's balance. This step-by-step approach not only helps in finding the solution but also reinforces the understanding of algebraic manipulations. Let's embark on this mathematical journey to unravel the value of x. This exercise will not only provide the answer but also enhance your problem-solving skills and understanding of mathematical principles. We will break down each step, explaining the rationale behind it, to ensure clarity and comprehension. By the end of this article, you will have a solid grasp of how to solve similar equations and a deeper appreciation for the elegance of algebra.

Understanding the Equation

The equation we aim to solve is 2.5(6x - 4) = 10 + 4(1.5 + 0.5x). This is a linear equation, characterized by the variable x being raised to the power of 1. To effectively solve this equation, we need to understand its structure and the operations involved. The equation consists of two sides, separated by an equals sign (=). The left side is 2.5(6x - 4), indicating that the expression 6x - 4 is multiplied by 2.5. The right side is 10 + 4(1.5 + 0.5x), showing that 10 is added to the product of 4 and the expression 1.5 + 0.5x. Our goal is to isolate x on one side of the equation to determine its value. This involves performing a series of algebraic operations while maintaining the equality. We will use the distributive property to remove parentheses, combine like terms to simplify both sides, and then apply inverse operations to get x by itself. The order of operations is crucial in this process. We must follow the correct sequence to ensure we arrive at the correct solution. Understanding the structure of the equation is the first step towards finding the value of x. By breaking down the equation into its components, we can approach the problem methodically and confidently. Each term and operation plays a vital role in the equation, and understanding their relationships is key to solving for x.

Step-by-Step Solution

To solve the equation 2.5(6x - 4) = 10 + 4(1.5 + 0.5x), we'll follow a step-by-step approach. First, we apply the distributive property to remove the parentheses on both sides of the equation. On the left side, we multiply 2.5 by both terms inside the parentheses: 2.5 * 6x = 15x and 2.5 * -4 = -10. So, the left side becomes 15x - 10. On the right side, we multiply 4 by both terms inside the parentheses: 4 * 1.5 = 6 and 4 * 0.5x = 2x. Thus, the right side becomes 10 + 6 + 2x, which simplifies to 16 + 2x. Now, our equation is 15x - 10 = 16 + 2x. Next, we want to gather all the x terms on one side and the constant terms on the other side. To do this, we subtract 2x from both sides: 15x - 2x - 10 = 16 + 2x - 2x, which simplifies to 13x - 10 = 16. Then, we add 10 to both sides to isolate the x term further: 13x - 10 + 10 = 16 + 10, which gives us 13x = 26. Finally, to solve for x, we divide both sides by 13: (13x) / 13 = 26 / 13, resulting in x = 2. This systematic approach ensures that we maintain the balance of the equation while isolating x. Each step is a logical progression, leading us closer to the solution. The meticulous application of algebraic principles is crucial for accurate problem-solving.

Detailed Explanation of Each Step

Let's break down each step of solving the equation 2.5(6x - 4) = 10 + 4(1.5 + 0.5x) in detail. The first step involves applying the distributive property. This property states that a(b + c) = ab + ac. On the left side of the equation, we distribute 2.5 across the terms inside the parentheses: 2.5 * 6x = 15x and 2.5 * -4 = -10. This transforms the left side into 15x - 10. Similarly, on the right side, we distribute 4 across the terms inside the parentheses: 4 * 1.5 = 6 and 4 * 0.5x = 2x. The right side now becomes 10 + 6 + 2x. We can further simplify this by combining the constants, 10 + 6, which equals 16. So, the right side is 16 + 2x. After the distribution and simplification, our equation is 15x - 10 = 16 + 2x. The next step is to collect like terms. This means grouping the terms with x on one side of the equation and the constant terms on the other side. To do this, we subtract 2x from both sides of the equation: 15x - 2x - 10 = 16 + 2x - 2x. This simplifies to 13x - 10 = 16. Next, we add 10 to both sides of the equation to isolate the x term: 13x - 10 + 10 = 16 + 10. This gives us 13x = 26. The final step is to solve for x by dividing both sides of the equation by the coefficient of x, which is 13: (13x) / 13 = 26 / 13. This results in x = 2. Each of these steps is crucial, and understanding the rationale behind them is key to mastering algebraic problem-solving. The distributive property allows us to eliminate parentheses, while collecting like terms simplifies the equation and makes it easier to isolate the variable. The final division isolates x, giving us the solution. This detailed explanation provides a thorough understanding of the process and reinforces the underlying mathematical principles.

Verifying the Solution

After finding a solution to an equation, it's essential to verify its correctness. This process involves substituting the obtained value back into the original equation and checking if both sides of the equation are equal. In our case, we found that x = 2 is the solution to the equation 2.5(6x - 4) = 10 + 4(1.5 + 0.5x). To verify this, we substitute x = 2 into the original equation: 2.5(6(2) - 4) = 10 + 4(1.5 + 0.5(2)). Now, we simplify both sides of the equation following the order of operations. On the left side, we first evaluate the expression inside the parentheses: 6(2) - 4 = 12 - 4 = 8. Then, we multiply this result by 2.5: 2.5 * 8 = 20. So, the left side of the equation becomes 20. On the right side, we first evaluate the expression inside the parentheses: 1.5 + 0.5(2) = 1.5 + 1 = 2.5. Then, we multiply this result by 4: 4 * 2.5 = 10. Finally, we add this to 10: 10 + 10 = 20. So, the right side of the equation also becomes 20. Since both sides of the equation are equal when x = 2, our solution is verified. This verification step is crucial because it helps to catch any potential errors made during the solving process. It ensures that the value we found for x satisfies the original equation, confirming the accuracy of our solution. This practice is a fundamental part of mathematical problem-solving and builds confidence in the correctness of the answer. By consistently verifying solutions, we reinforce our understanding of the equation and the algebraic manipulations performed.

Conclusion

In this article, we have successfully solved the linear equation 2.5(6x - 4) = 10 + 4(1.5 + 0.5x) for x. We began by understanding the structure of the equation and identifying the operations involved. Then, we applied a systematic approach, using the distributive property to remove parentheses, combining like terms to simplify the equation, and employing inverse operations to isolate x. Through these steps, we determined that x = 2 is the solution. Furthermore, we emphasized the importance of verifying the solution by substituting x = 2 back into the original equation. This verification step confirmed that our solution is correct, as both sides of the equation were equal. The process of solving linear equations is a fundamental skill in mathematics, with applications across various disciplines. The ability to manipulate equations and isolate variables is crucial for problem-solving in physics, engineering, economics, and computer science, among other fields. This article not only provided a step-by-step solution to a specific equation but also highlighted the underlying principles and techniques involved in solving linear equations in general. By mastering these principles, you can confidently tackle a wide range of mathematical problems. The key takeaways from this exercise include the importance of the distributive property, the need to combine like terms, the use of inverse operations, and the significance of verifying solutions. These skills form the foundation for more advanced mathematical concepts and are essential for success in quantitative fields. Practice and a solid understanding of these concepts will empower you to solve increasingly complex problems with accuracy and efficiency.