Solving For X In The Equation 3x - 4y = 65 Given Y = 4

Introduction

In this article, we will delve into the process of solving for the value of x in the linear equation 3x - 4y = 65, given that y = 4. Linear equations are fundamental mathematical expressions that describe relationships between variables, and mastering the techniques to solve them is crucial in various fields, including algebra, calculus, and real-world problem-solving. This exploration aims to provide a comprehensive understanding of the steps involved, ensuring clarity and accuracy in the solution. We will begin by substituting the given value of y into the equation, simplifying the expression, and then isolating x to determine its value. This process not only demonstrates a specific solution but also reinforces the general principles of algebraic manipulation applicable to a wide range of problems.

Understanding Linear Equations

At its core, a linear equation represents a straight line on a graph, illustrating a direct relationship between two or more variables. In the equation 3x - 4y = 65, we have two variables, x and y, and our goal is to find the value of x when y is known. The beauty of linear equations lies in their simplicity and predictability; a change in one variable results in a proportional change in the other. This property makes them incredibly useful for modeling real-world scenarios, from calculating distances and speeds to determining financial growth and decay. The ability to solve these equations efficiently is a cornerstone of mathematical literacy, allowing us to make informed decisions and predictions based on quantitative data. Moreover, understanding the underlying principles of linear equations provides a solid foundation for tackling more complex mathematical concepts, such as systems of equations, inequalities, and advanced algebraic functions. Therefore, mastering the techniques for solving linear equations is not just an academic exercise but a practical skill with far-reaching applications in various aspects of life and work.

Step-by-Step Solution

Substituting the Value of y

The initial step in solving for x is to substitute the given value of y, which is 4, into the equation. This substitution allows us to reduce the equation to a single variable, making it easier to isolate and solve for x. Replacing y with 4 in the equation 3x - 4y = 65, we get:

3x - 4(4) = 65

This substitution is a crucial step as it simplifies the equation by eliminating one variable, setting the stage for further algebraic manipulation. The process of substitution is a fundamental technique in solving equations, applicable not only to linear equations but also to more complex systems and functions. It involves replacing a variable with its known value or an equivalent expression, allowing us to reduce the complexity of the equation and focus on the unknown variable we aim to find. In this case, substituting y = 4 transforms the equation into a simpler form that can be readily solved for x. This step highlights the importance of careful substitution and attention to detail, as any error in this stage can propagate through the rest of the solution process.

Simplifying the Equation

After substituting the value of y, the next step involves simplifying the equation. This means performing any arithmetic operations to reduce the equation to its simplest form. In our case, we need to multiply -4 by 4:

3x - 16 = 65

Simplifying equations is a critical skill in algebra, as it allows us to work with manageable expressions and accurately isolate the variable we are trying to solve for. This step often involves combining like terms, distributing coefficients, and performing arithmetic operations such as addition, subtraction, multiplication, and division. The goal is to transform the equation into a form where the variable is isolated on one side and a constant term is on the other. In this specific example, simplifying the equation involves multiplying -4 by 4 to get -16, which is then combined with the other terms in the equation. This simplification not only makes the equation easier to work with but also reduces the likelihood of errors in subsequent steps. The ability to simplify equations effectively is essential for solving a wide range of mathematical problems and is a fundamental skill in algebraic manipulation.

Isolating x

To isolate x, we need to get it by itself on one side of the equation. This involves performing inverse operations to undo any operations that are being applied to x. In our equation, 3x - 16 = 65, x is being multiplied by 3 and then 16 is being subtracted. To isolate x, we first need to undo the subtraction by adding 16 to both sides of the equation:

3x - 16 + 16 = 65 + 16

3x = 81

Next, we need to undo the multiplication by 3. We do this by dividing both sides of the equation by 3:

(3x)/3 = 81/3

x = 27

The process of isolating a variable is a cornerstone of solving algebraic equations. It involves strategically applying inverse operations to both sides of the equation to gradually peel away the terms and coefficients surrounding the variable until it stands alone. This technique is based on the fundamental principle that performing the same operation on both sides of an equation maintains the equality. In this case, we first added 16 to both sides to eliminate the -16 term, and then we divided both sides by 3 to eliminate the coefficient of x. Each step is carefully chosen to move closer to isolating x while preserving the balance of the equation. Mastering the art of isolating variables is essential for solving a wide range of algebraic problems and forms the foundation for more advanced mathematical concepts.

Solution

Therefore, the value of x in the equation 3x - 4y = 65, when y = 4, is 27.

Verification

To ensure the accuracy of our solution, it's always a good practice to verify the result. This involves substituting the calculated value of x back into the original equation, along with the given value of y, and checking if the equation holds true. Substituting x = 27 and y = 4 into the equation 3x - 4y = 65, we get:

3(27) - 4(4) = 65

81 - 16 = 65

65 = 65

The equation holds true, confirming that our solution x = 27 is correct. Verification is a critical step in the problem-solving process, as it provides a check for errors and ensures the reliability of the solution. It involves retracing the steps of the solution and verifying that each step is logically sound and arithmetically correct. This process not only confirms the accuracy of the answer but also reinforces the understanding of the underlying mathematical concepts. In this case, verifying the solution by substituting the values of x and y back into the original equation provides a high level of confidence in the correctness of the result.

Conclusion

In this article, we successfully solved for x in the equation 3x - 4y = 65, given y = 4. By following a step-by-step approach, we substituted the value of y, simplified the equation, isolated x, and verified our solution. This exercise not only demonstrates the application of algebraic principles but also reinforces the importance of accuracy and verification in mathematical problem-solving. The ability to solve linear equations is a fundamental skill in mathematics, with applications in various fields. Mastering these techniques allows us to tackle more complex problems and develop a deeper understanding of mathematical concepts. The process of solving for x in this equation highlights the power of algebraic manipulation and the importance of following a systematic approach to problem-solving. The skills acquired in this exercise can be applied to a wide range of mathematical problems, making it a valuable learning experience for anyone studying algebra or related fields.