Solving For X In The Equation 4x + 1/3 = (1/3)x + 4
Introduction
In the realm of mathematics, solving equations is a fundamental skill. This article will guide you through the process of finding the value of x in the equation 4x + 1/3 = (1/3)x + 4. This equation involves linear expressions, and we will utilize algebraic manipulations to isolate x and determine its value. Understanding how to solve such equations is crucial for various mathematical and real-world applications. The principles we'll cover here are applicable to a wide range of linear equations, making this a valuable exercise for anyone looking to strengthen their algebra skills. We'll break down each step in detail, ensuring that the reasoning behind each manipulation is clear. This approach will not only help you solve this specific equation but also provide you with a solid foundation for tackling more complex problems in the future. So, let's dive into the process of solving for x and unlock the solution to this equation.
Step-by-Step Solution
1. Eliminate Fractions
To simplify the equation, the first step is often to eliminate fractions. This can be achieved by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the denominator is 3. Multiplying both sides of the equation 4x + 1/3 = (1/3)x + 4 by 3, we get:
3 * (4x + 1/3) = 3 * ((1/3)x + 4)
Distributing the 3 on both sides:
3 * 4x + 3 * (1/3) = 3 * (1/3)x + 3 * 4
This simplifies to:
12x + 1 = x + 12
Eliminating fractions makes the equation easier to work with, as we now have a linear equation without any fractional coefficients. This is a crucial step in solving for x, as it sets the stage for further algebraic manipulations. By removing the fractions, we can proceed with isolating x using standard techniques such as combining like terms and performing inverse operations. The goal here is to transform the equation into a form where x is on one side and a constant value is on the other, allowing us to directly read off the solution. This initial step of eliminating fractions is a common strategy in solving various types of equations, highlighting its importance in algebraic problem-solving. So, with the fractions out of the way, let's move on to the next step in our journey to find the value of x.
2. Combine Like Terms
Now that we have the equation 12x + 1 = x + 12, the next step is to combine like terms. This involves grouping the terms with x on one side of the equation and the constant terms on the other side. To do this, we can subtract x from both sides:
12x + 1 - x = x + 12 - x
This simplifies to:
11x + 1 = 12
Next, we subtract 1 from both sides to isolate the term with x:
11x + 1 - 1 = 12 - 1
Which gives us:
11x = 11
Combining like terms is a fundamental algebraic technique that simplifies the equation and brings us closer to isolating x. By moving all the x terms to one side and the constants to the other, we create a clearer path towards solving for the unknown variable. This step is crucial because it reduces the complexity of the equation, making it easier to apply the final steps of solving. The process of combining like terms involves careful application of algebraic principles, ensuring that we maintain the equality of both sides of the equation. This technique is not only useful in solving linear equations but also in simplifying more complex expressions and equations in various branches of mathematics. So, having successfully combined like terms, we are now just one step away from finding the value of x. Let's proceed to the final step and complete our solution.
3. Isolate x
We have arrived at the equation 11x = 11. To isolate x, we need to divide both sides of the equation by the coefficient of x, which is 11:
(11x) / 11 = 11 / 11
This simplifies to:
x = 1
Therefore, the value of x that satisfies the equation 4x + 1/3 = (1/3)x + 4 is 1. Isolating x is the ultimate goal in solving for an unknown variable. This step involves performing the inverse operation of the coefficient of x, effectively undoing the multiplication and leaving x by itself on one side of the equation. The result is a direct solution for x, providing the numerical value that makes the equation true. This final step is the culmination of all the previous algebraic manipulations, highlighting the importance of each step in the overall process. Isolating x is a fundamental skill in algebra and is essential for solving a wide variety of equations and problems. With x now isolated, we have successfully found the solution to the equation. Let's summarize our findings and conclude our exploration.
Conclusion
By following these steps, we have successfully found that x = 1 is the solution to the equation 4x + 1/3 = (1/3)x + 4. This process demonstrates the systematic approach to solving linear equations, which involves eliminating fractions, combining like terms, and isolating the variable. These skills are essential for success in algebra and beyond. The ability to solve equations is a cornerstone of mathematical proficiency, enabling us to tackle a wide range of problems in various fields. The steps we've outlined here provide a clear framework for solving linear equations, emphasizing the importance of each manipulation in arriving at the correct solution. Mastering these techniques not only enhances our problem-solving abilities but also deepens our understanding of mathematical principles. So, with the solution to this equation in hand, we can appreciate the power of algebraic methods and their applicability in unraveling mathematical puzzles. Keep practicing and applying these skills, and you'll find yourself well-equipped to tackle more complex challenges in the world of mathematics.
