Solving X/3 = X/4 - 4 A Step-by-Step Guide

In the realm of mathematics, equations form the bedrock of problem-solving. Among these, linear equations hold a prominent position, serving as fundamental tools in various disciplines. This article delves into the intricacies of solving the linear equation $\frac{x}{3} = \frac{x}{4} - 4$, providing a step-by-step guide to unravel its solution.

Understanding the Equation

Before embarking on the solution process, it's crucial to understand the equation at hand. The equation $\frac{x}{3} = \frac{x}{4} - 4$ represents a relationship between an unknown variable, denoted as 'x', and constant values. Our objective is to isolate 'x' on one side of the equation, thereby determining its numerical value. This equation, in particular, falls under the category of linear equations, characterized by the variable 'x' being raised to the power of 1. Linear equations are ubiquitous in mathematics and its applications, appearing in fields ranging from physics and engineering to economics and computer science.

The equation $\frac{x}{3} = \frac{x}{4} - 4$ presents a scenario where the unknown variable 'x' appears on both sides of the equation. This necessitates a strategic approach to isolate 'x' and arrive at a solution. The presence of fractions, namely $\frac{x}{3}$ and $\frac{x}{4}$, adds a layer of complexity, requiring us to employ techniques to eliminate these fractions and simplify the equation. To effectively tackle this equation, we'll leverage the fundamental principles of algebraic manipulation, ensuring that we maintain the equality on both sides of the equation throughout the solution process. This involves performing operations such as addition, subtraction, multiplication, and division on both sides of the equation, adhering to the golden rule of equation solving: whatever operation is performed on one side must also be performed on the other. By meticulously applying these principles, we can systematically isolate 'x' and unveil its value.

Step 1: Eliminating Fractions

The presence of fractions often complicates the process of solving equations. To circumvent this, our initial step involves eliminating the fractions from the equation. In the equation $\frac{x}{3} = \frac{x}{4} - 4$, we have two fractions: $\frac{x}{3}$ and $\frac{x}{4}$. To eliminate these fractions, we seek a common multiple of the denominators, 3 and 4. The least common multiple (LCM) of 3 and 4 is 12. Multiplying both sides of the equation by 12 will effectively clear the fractions.

Multiplying both sides of the equation by 12, we get:

$$12 * (\frac{x}{3}) = 12 * (\frac{x}{4} - 4)$$

This simplifies to:

$$4x = 3x - 48$$

Now, we have successfully eliminated the fractions, transforming the equation into a simpler form that is easier to manipulate. This step is crucial as it allows us to work with whole numbers, making the subsequent steps more straightforward. The equation $4x = 3x - 48$ no longer contains any fractions, making it more amenable to algebraic manipulation. We can now proceed to isolate the variable 'x' by employing techniques such as combining like terms and performing inverse operations. By eliminating the fractions, we have laid a solid foundation for solving the equation and determining the value of 'x'. This step underscores the importance of recognizing and addressing potential complications, such as fractions, in order to streamline the solution process.

Step 2: Isolating the Variable

With the fractions eliminated, our next objective is to isolate the variable 'x' on one side of the equation. In the equation $4x = 3x - 48$, we have 'x' terms on both sides. To isolate 'x', we need to gather all the 'x' terms on one side and the constant terms on the other. This can be achieved by performing algebraic manipulations that maintain the equality of the equation. Subtracting 3x from both sides is a strategic move that will consolidate the 'x' terms on the left side of the equation.

Subtracting 3x from both sides, we get:

$$4x - 3x = 3x - 48 - 3x$$

This simplifies to:

$$x = -48$$

By subtracting 3x from both sides, we have successfully isolated 'x' on the left side of the equation. The equation now reads $x = -48$, which directly reveals the value of 'x'. This step showcases the power of algebraic manipulation in rearranging equations to unveil the unknown variable. The ability to isolate variables is a fundamental skill in algebra and is essential for solving a wide range of equations. The process of isolating 'x' involved strategic application of inverse operations, ensuring that the equality of the equation was maintained throughout. The result, $x = -48$, provides the solution to the original equation, indicating the numerical value that satisfies the given relationship.

Step 3: Verifying the Solution

After obtaining a solution, it's prudent to verify the solution to ensure its accuracy. This step involves substituting the obtained value of 'x' back into the original equation and checking if the equation holds true. In our case, we found that $x = -48$. Substituting this value into the original equation, $\frac{x}{3} = \frac{x}{4} - 4$, allows us to confirm whether it satisfies the equation.

Substituting $x = -48$ into the original equation, we get:

$$\frac{-48}{3} = \frac{-48}{4} - 4$$

Simplifying both sides, we have:

$$-16 = -12 - 4$$

$$-16 = -16$$

The equation holds true, confirming that $x = -48$ is indeed the correct solution. This verification step is crucial as it helps to catch any potential errors made during the solution process. It provides confidence in the accuracy of the solution and ensures that it satisfies the given equation. Verification is a cornerstone of problem-solving in mathematics and is applicable across various mathematical domains. By substituting the solution back into the original equation, we are essentially performing a check to ensure that the left-hand side of the equation equals the right-hand side. If the two sides are equal, then the solution is verified; otherwise, it indicates that an error may have occurred during the solution process. In this case, the verification confirms that our solution, $x = -48$, is accurate and satisfies the equation $\frac{x}{3} = \frac{x}{4} - 4$.

Conclusion

In this article, we have meticulously solved the linear equation $\frac{x}{3} = \frac{x}{4} - 4$, demonstrating a step-by-step approach that can be applied to a wide range of equations. We began by understanding the equation and identifying the key challenges, such as the presence of fractions. We then systematically addressed these challenges by eliminating the fractions and isolating the variable 'x'. Finally, we verified our solution to ensure its accuracy. The solution to the equation is $x = -48$.

The process of solving equations is a fundamental skill in mathematics, and this article has provided a comprehensive guide to tackling linear equations. The steps outlined here, including eliminating fractions, isolating variables, and verifying solutions, form the foundation for solving more complex equations. By mastering these techniques, individuals can confidently approach mathematical problems and develop their problem-solving abilities. The equation $\frac{x}{3} = \frac{x}{4} - 4$ serves as a valuable example of how algebraic manipulation can be used to unveil the unknown. The solution, $x = -48$, represents the value that satisfies the given relationship, and the verification process reinforces the accuracy of the solution. This article underscores the importance of a systematic approach to equation solving, emphasizing the need for careful manipulation, attention to detail, and verification of results. By following these principles, individuals can enhance their mathematical proficiency and confidently tackle equations of varying complexity.