Solving For K In The Equation 8k + 2m = 3m + K

JU07/09/2025 08, 2025 by THE IDEN 47 views

In this comprehensive guide, we will delve into the step-by-step process of solving for the variable k in the algebraic equation 8k + 2m = 3m + k. This equation involves two variables, k and m, and our primary goal is to isolate k on one side of the equation to determine its value in terms of m. Understanding how to solve such equations is a fundamental skill in algebra and has widespread applications in various fields, including physics, engineering, and economics. Whether you're a student looking to solidify your algebraic skills or simply someone interested in mathematical problem-solving, this detailed explanation will provide you with a clear and concise approach to tackling similar problems.

Algebraic equations are the backbone of mathematical modeling and problem-solving. They allow us to represent relationships between different quantities and to find unknown values. Equations like 8k + 2m = 3m + k are known as linear equations, where the highest power of the variables is 1. Solving for a specific variable in a linear equation involves using algebraic manipulations to isolate that variable on one side of the equation. This process typically involves combining like terms, adding or subtracting terms from both sides of the equation, and multiplying or dividing both sides by a constant. The key principle is to maintain the equality of the equation while simplifying it to a form where the desired variable is expressed in terms of the other variables or constants.

Before we dive into the step-by-step solution, it's crucial to understand the underlying principles of algebraic manipulation. The golden rule of equation solving is that any operation performed on one side of the equation must also be performed on the other side to maintain balance. This ensures that the equality remains valid throughout the process. For instance, if we subtract a term from one side of the equation, we must subtract the same term from the other side. Similarly, if we multiply one side of the equation by a constant, we must multiply the other side by the same constant. These principles are the foundation of algebraic manipulation and allow us to systematically simplify equations to isolate the variable of interest. Mastering these techniques is essential for success in algebra and other mathematical disciplines.

To effectively solve for k in the equation 8k + 2m = 3m + k, we will follow a series of well-defined algebraic steps. Each step is designed to bring us closer to isolating k on one side of the equation. By meticulously applying these steps, we can ensure an accurate and efficient solution. Let's break down the process:

Step 1: Combine Like Terms

The first step in solving for k is to gather all terms containing k on one side of the equation and all terms containing m on the other side. This involves rearranging the equation to group similar terms together. In our equation, we have 8k on the left side and k on the right side. To combine these terms, we need to eliminate the k term from the right side. We can achieve this by subtracting k from both sides of the equation.

Original Equation: 8k + 2m = 3m + k
Subtract k from both sides: 8k + 2m - k = 3m + k - k
Simplified Equation: 7k + 2m = 3m

Now, we have all the k terms on the left side. Next, we need to move the terms containing m to the right side. We have 2m on the left side and 3m on the right side. To isolate the k term further, we will subtract 2m from both sides of the equation.

Current Equation: 7k + 2m = 3m
Subtract 2m from both sides: 7k + 2m - 2m = 3m - 2m
Simplified Equation: 7k = m

At this point, we have successfully grouped the k terms on the left side and the m terms on the right side. This step is crucial because it simplifies the equation and brings us closer to isolating k.

Step 2: Isolate k

With the equation now in the form 7k = m, our next and final step is to isolate k completely. This means we need to get k by itself on one side of the equation. Since k is being multiplied by 7, we can undo this multiplication by dividing both sides of the equation by 7. This is a fundamental algebraic operation that preserves the equality of the equation while isolating the desired variable.

Current Equation: 7k = m
Divide both sides by 7: (7k) / 7 = m / 7
Simplified Equation: k = m / 7

By dividing both sides by 7, we have successfully isolated k. The equation now expresses k in terms of m. This means that the value of k is directly proportional to the value of m, with a constant of proportionality of 1/7. This is the solution to our original equation, and it provides a clear relationship between k and m.

After following the steps of combining like terms and isolating k, we have arrived at the solution:

k = m / 7

This equation tells us that the value of k is equal to the value of m divided by 7. This is the final solution to the problem, and it represents the value of k that satisfies the original equation 8k + 2m = 3m + k. It's important to note that this solution expresses k in terms of m, meaning that the value of k depends on the value of m. If we were given a specific value for m, we could substitute it into this equation to find the corresponding value of k. However, in this case, we have solved for k in its most general form, expressing it as a function of m.

In conclusion, we have successfully solved for k in the equation 8k + 2m = 3m + k. By systematically combining like terms and isolating k, we arrived at the solution k = m / 7. This process demonstrates the fundamental principles of algebraic manipulation and highlights the importance of maintaining balance in equations. Solving for variables in algebraic equations is a core skill in mathematics and has wide-ranging applications in various fields.

This step-by-step guide provides a clear and concise approach to solving similar equations. By understanding the underlying principles and practicing these techniques, you can confidently tackle a variety of algebraic problems. Remember, the key to success in algebra is to break down complex problems into smaller, manageable steps and to apply the rules of algebra consistently. With practice and patience, you can master these skills and excel in mathematics.

Algebraic problem-solving is not just about finding the right answer; it's also about developing critical thinking and logical reasoning skills. By working through problems like this, you are honing your ability to analyze information, identify patterns, and apply appropriate strategies. These skills are valuable not only in mathematics but also in many other areas of life. So, continue to practice and explore algebraic concepts, and you will find that your problem-solving abilities will grow stronger over time.

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