Finding The Value Of X In The Equation 3x - 4y = 65

In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of finding the value of the variable x in the equation 3x - 4y = 65, given that y = 4. We will not only provide the solution but also offer a step-by-step explanation, ensuring a clear understanding of the underlying principles. Whether you're a student grappling with algebraic equations or simply seeking to enhance your mathematical prowess, this guide will equip you with the knowledge and confidence to tackle similar problems.

Deciphering the Equation: A Step-by-Step Approach

The heart of this mathematical exploration lies in the equation 3x - 4y = 65. This equation represents a linear relationship between two variables, x and y. Our mission is to isolate x and determine its value when y is given as 4. To achieve this, we will employ a series of algebraic manipulations, meticulously unraveling the equation to reveal the hidden value of x.

1. The Power of Substitution: Replacing y with its Numerical Value

The first step in our journey involves the substitution of y with its numerical value, which is 4. This substitution is a cornerstone of equation solving, allowing us to transform the equation into a simpler form with only one unknown variable. By replacing y with 4, we effectively eliminate one variable, paving the way for us to isolate x. This process transforms the equation as follows:

3x - 4(4) = 65

This seemingly simple substitution is a pivotal moment in our quest, bridging the gap between a two-variable equation and a solvable one-variable equation.

2. Simplifying the Equation: Unveiling the Arithmetic Core

With the substitution complete, our next task is to simplify the equation by performing the arithmetic operations. This step involves the multiplication of -4 and 4, resulting in -16. This simplification process is akin to peeling away the layers of an onion, revealing the core mathematical structure of the equation. The equation now takes the form:

3x - 16 = 65

This simplified form brings us closer to our goal of isolating x. The subtraction of 16 from 3x is a key element that we must address to further our progress.

3. Isolating the Term with x: The Art of Adding to Both Sides

To isolate the term containing x, which is 3x, we employ the fundamental principle of algebraic manipulation: performing the same operation on both sides of the equation. In this case, we add 16 to both sides of the equation. This strategic move cancels out the -16 on the left side, leaving us with 3x standing alone. The equation transforms as follows:

3x - 16 + 16 = 65 + 16

This addition is a critical step in our isolation strategy, effectively removing the constant term from the left side of the equation. The result of this operation is:

3x = 81

We are now on the verge of discovering the value of x. The equation 3x = 81 presents a clear path forward, requiring only one more step to unveil the solution.

4. Unveiling x: The Power of Division

The final step in our algebraic journey involves isolating x by dividing both sides of the equation by 3. This division is the culmination of our efforts, effectively undoing the multiplication of x by 3. By dividing both sides by 3, we arrive at the value of x. The equation transforms as follows:

3x / 3 = 81 / 3

This division is the ultimate key to unlocking the value of x. Performing the division yields the solution:

x = 27

Thus, we have successfully determined the value of x in the equation 3x - 4y = 65 when y = 4. The solution is x = 27.

Validating the Solution: Ensuring Accuracy and Precision

With the solution x = 27 in hand, it is crucial to validate our answer. This validation process involves substituting the values of x and y back into the original equation to ensure that the equation holds true. This step is a cornerstone of mathematical rigor, confirming the accuracy of our calculations and the validity of our solution. Substituting x = 27 and y = 4 into the original equation, we get:

3(27) - 4(4) = 65

Simplifying the equation, we have:

81 - 16 = 65

This simplifies further to:

65 = 65

The equation holds true, confirming that our solution x = 27 is indeed correct. This validation process not only provides assurance but also reinforces our understanding of the equation and the solution process.

Conclusion: Mastering the Art of Equation Solving

In this exploration, we have successfully navigated the equation 3x - 4y = 65, unraveling the value of x when y is 4. Through a series of strategic substitutions, simplifications, and algebraic manipulations, we arrived at the solution x = 27. The journey involved substituting y with its numerical value, simplifying the equation, isolating the term containing x, and finally, unveiling x through division. Furthermore, we validated our solution, ensuring its accuracy and precision.

This exercise not only provides the answer to a specific problem but also illuminates the broader principles of equation solving. The techniques employed here, such as substitution, simplification, and isolating variables, are fundamental tools in the mathematician's arsenal. By mastering these techniques, you can confidently tackle a wide range of algebraic equations, unlocking the power of mathematics to solve problems in various domains.

The ability to solve equations is not merely an academic pursuit; it is a life skill that transcends the classroom. From financial planning to scientific research, equation solving plays a crucial role in decision-making and problem-solving across diverse fields. By honing your equation-solving skills, you empower yourself to navigate the complexities of the world with greater clarity and precision. Embrace the challenge of equations, and you will unlock a world of possibilities.

This comprehensive guide has equipped you with the knowledge and confidence to tackle similar problems. Remember, the key to mathematical mastery lies in practice and perseverance. Continue to explore, experiment, and refine your skills, and you will undoubtedly excel in the realm of equation solving. So, go forth and conquer the world of equations!

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