Solving For X In 3x - 4y = 65 When Y Is 4

Finding the value of x in a given equation is a fundamental concept in algebra. In this article, we will explore how to solve for x in the equation 3x - 4y = 65, given that y = 4. This problem demonstrates a typical algebraic manipulation where we substitute a known value and then isolate the variable we want to find. Understanding these steps is crucial for anyone studying algebra, as it lays the groundwork for more complex problem-solving scenarios. We will break down each step clearly, ensuring that you can follow along and apply the same method to similar problems. Let's dive into the solution to uncover the correct value of x.

Understanding the Problem

Before we jump into the solution, it's essential to understand the problem. We are given the equation 3x - 4y = 65, which is a linear equation with two variables, x and y. Our goal is to find the value of x. We are also given that y = 4, which is the key to solving this problem. By substituting the value of y into the equation, we can reduce it to an equation with only one variable, x, which we can then solve using basic algebraic techniques. This process of substitution is a common strategy in algebra and is used extensively in various mathematical contexts. Make sure you grasp the concept of substituting known values to simplify equations, as it is a foundational skill for more advanced topics. Now, let's proceed to the steps involved in solving for x.

Step-by-Step Solution

To find the value of x in the equation 3x - 4y = 65 when y = 4, we'll follow these steps:

1. Substitute the Value of y

The first step is to substitute the given value of y into the equation. We are given that y = 4, so we replace y with 4 in the equation:

3x - 4(4) = 65

This substitution simplifies the equation by eliminating one variable, making it easier to solve for x. Substitution is a fundamental technique in algebra and is used to reduce complex equations into simpler forms. By replacing a variable with its known value, we create an equation with a single unknown, which can then be solved using basic algebraic operations.

2. Simplify the Equation

Next, we simplify the equation by performing the multiplication:

3x - 16 = 65

Here, we multiplied -4 by 4 to get -16. Simplifying equations is a crucial step in solving for variables. By performing arithmetic operations and combining like terms, we make the equation more manageable and easier to solve. In this case, simplifying the equation involves multiplying constants and preparing the equation for the next step, which is to isolate the variable x. This process of simplification is a cornerstone of algebraic manipulation and is essential for solving a wide range of mathematical problems.

3. Isolate the Term with x

To isolate the term with x, we need to get rid of the -16 on the left side of the equation. We do this by adding 16 to both sides of the equation:

3x - 16 + 16 = 65 + 16

This step is based on the principle that adding the same value to both sides of an equation maintains the equality. By adding 16 to both sides, we cancel out the -16 on the left side, leaving us with the term containing x. Isolating the variable term is a critical step in solving equations because it brings us closer to finding the value of the variable. This process is a fundamental algebraic technique that is used in solving various types of equations.

4. Further Simplification

Simplifying further, we get:

3x = 81

Here, we added 65 and 16 to get 81. This simplification step is essential to making the equation easier to solve. By performing the addition, we isolate the term with x on one side of the equation and a constant on the other side. This step sets up the final step in solving for x, which is to divide both sides of the equation by the coefficient of x. Simplifying the equation at each step ensures that the calculations remain manageable and the solution is reached efficiently.

5. Solve for x

Finally, to solve for x, we divide both sides of the equation by 3:

3x / 3 = 81 / 3

This division isolates x on the left side of the equation. Dividing both sides by the coefficient of x is the final step in solving for the variable. This step is based on the principle that dividing both sides of an equation by the same non-zero value maintains the equality. By performing this division, we find the value of x that satisfies the original equation. This process is a fundamental algebraic technique and is essential for solving various types of equations.

6. Calculate the Value of x

Calculating the division, we find:

x = 27

Thus, the value of x that satisfies the equation 3x - 4y = 65 when y = 4 is 27. This is the final step in solving the problem. After performing all the necessary algebraic manipulations, we have successfully isolated x and found its value. This solution demonstrates the step-by-step process of solving a linear equation with one variable, which is a fundamental skill in algebra. Understanding and mastering this process is crucial for tackling more complex mathematical problems.

Detailed Explanation of Each Step

To ensure clarity, let's delve deeper into each step of the solution process. Understanding the rationale behind each step is crucial for mastering algebraic problem-solving.

1. Substitute the Value of y

The substitution step is fundamental in solving equations with multiple variables. In our case, the equation 3x - 4y = 65 involves two variables, x and y. However, we are given that y = 4. By substituting this value into the equation, we eliminate the variable y and transform the equation into one with a single variable, x. This is a common technique in algebra because it simplifies the equation and makes it easier to solve. The act of substitution allows us to work with a more manageable equation, focusing solely on finding the value of x. This step is not only crucial for solving this particular problem but also serves as a foundational skill for tackling more complex algebraic problems involving multiple variables.

2. Simplify the Equation

Simplifying the equation after substitution is a critical step in isolating the variable x. In our case, after substituting y = 4 into the equation, we have 3x - 4(4) = 65. The next step is to perform the multiplication, which yields 3x - 16 = 65. This simplification involves basic arithmetic operations and is essential for making the equation more manageable. Simplifying the equation reduces the number of terms and operations, making it easier to see the next steps required to solve for x. This process not only prepares the equation for further manipulation but also reduces the likelihood of errors in subsequent calculations. Simplifying equations is a fundamental skill in algebra and is crucial for efficient problem-solving.

3. Isolate the Term with x

Isolating the term with x is a key step in solving for the variable. In our equation, 3x - 16 = 65, we want to isolate the term 3x on one side of the equation. To do this, we add 16 to both sides of the equation. Adding the same value to both sides maintains the equality and allows us to eliminate the -16 on the left side. This step results in the equation 3x = 65 + 16. Isolating the term with x is crucial because it brings us closer to finding the value of x. This technique is based on the fundamental principle that performing the same operation on both sides of an equation preserves the balance, which is essential for solving algebraic equations.

4. Further Simplification

Further simplifying the equation is an essential step to prepare for the final solution. After isolating the term with x, we have 3x = 65 + 16. The next step is to perform the addition on the right side of the equation, which yields 3x = 81. This simplification combines the constant terms, making the equation more straightforward. Simplifying the equation reduces the complexity and makes it easier to see the final step required to solve for x. This step is part of the broader strategy of algebraic manipulation, where each step aims to reduce the equation to its simplest form, ultimately leading to the solution. Accurate simplification is vital for avoiding errors and ensuring the correct value of x is found.

5. Solve for x

Solving for x involves isolating the variable by performing the inverse operation. In our equation, 3x = 81, x is multiplied by 3. To isolate x, we divide both sides of the equation by 3. This division cancels out the multiplication by 3 on the left side, leaving x by itself. This step results in the equation x = 81 / 3. Dividing both sides by the same value maintains the equality and is a fundamental algebraic technique. Isolating x is the ultimate goal of solving the equation, and this step brings us to the final calculation needed to find the value of x. The process of isolating the variable through inverse operations is a core concept in algebra.

6. Calculate the Value of x

Calculating the value of x is the final step in solving the equation. After isolating x, we have x = 81 / 3. Performing the division, we find that x = 27. This is the solution to the equation 3x - 4y = 65 when y = 4. The calculation provides the numerical value of x that satisfies the given conditions. This step concludes the algebraic manipulation and provides a concrete answer to the problem. Verifying the solution by substituting it back into the original equation is a good practice to ensure accuracy. The final calculation is the culmination of all the previous steps, demonstrating the power of algebraic techniques in problem-solving.

Conclusion

In conclusion, by following the step-by-step process of substitution, simplification, isolation, and calculation, we found that the value of x in the equation 3x - 4y = 65, when y = 4, is 27. This exercise demonstrates the importance of understanding and applying fundamental algebraic techniques. Each step, from substituting the known value of y to isolating and solving for x, is crucial in reaching the correct solution. Mastering these techniques not only helps in solving equations but also lays a strong foundation for more advanced mathematical concepts. Remember to always simplify equations step by step and to verify your solutions whenever possible. With practice, solving algebraic equations becomes second nature, opening doors to further mathematical exploration and problem-solving.