Solving For K In The Equation 3 - 3k + 7k = 5b
Introduction
In this comprehensive guide, we will delve into the process of solving the equation 3 - 3k + 7k = 5b for the variable k. This equation is a fundamental example of a linear equation, and mastering the steps to solve it is crucial for building a strong foundation in algebra. Whether you're a student learning the basics or someone looking to refresh your algebra skills, this article provides a detailed, step-by-step approach to solving for k, ensuring clarity and understanding every step of the way. By the end of this article, you'll not only know how to solve this particular equation but also gain a broader understanding of how to manipulate and solve similar algebraic problems. Our goal is to break down the solution into manageable parts, making it easy to follow along and replicate on your own.
Understanding the Equation
Before we jump into solving for k, it's essential to understand the structure of the equation we're working with. The equation 3 - 3k + 7k = 5b is a linear equation, which means that the highest power of the variable k is 1. Linear equations are characterized by their straightforward relationships between variables, making them relatively simple to solve compared to higher-degree equations. In this equation, we have terms involving k, a constant term (3), and a term involving another variable, b. The presence of b means that our solution for k will likely be expressed in terms of b. To solve for k, we need to isolate k on one side of the equation. This involves combining like terms, moving terms across the equals sign, and performing algebraic operations that maintain the equation's balance. By understanding the nature of the equation and the goal of isolating k, we set the stage for a clear and methodical solution process. Each step we take will bring us closer to expressing k in terms of b, providing a general solution that works for any value of b.
Step-by-Step Solution
To effectively solve the equation 3 - 3k + 7k = 5b for k, we will follow a systematic, step-by-step approach. This method ensures that each operation is performed correctly, leading to an accurate solution. By breaking down the process into smaller, manageable steps, we can minimize errors and gain a deeper understanding of the algebraic manipulations involved. Let's begin:
Step 1: Combine Like Terms
The first step in solving the equation is to combine the like terms on the left side. In this case, the like terms are the k terms: -3k and +7k. Combining these terms gives us:
-3k + 7k = 4k
So, the equation now becomes:
3 + 4k = 5b
Combining like terms simplifies the equation, making it easier to isolate the variable k. This step reduces the complexity of the equation, allowing us to proceed with the next steps more efficiently. By simplifying the equation, we maintain its balance and prepare it for further algebraic manipulation. This is a crucial first step in solving any algebraic equation, as it sets the stage for the subsequent steps and helps prevent errors.
Step 2: Isolate the Term with k
Next, we need to isolate the term containing k on one side of the equation. Currently, we have the equation:
3 + 4k = 5b
To isolate the term 4k, we need to eliminate the constant term, 3, from the left side. We can do this by subtracting 3 from both sides of the equation. This maintains the balance of the equation and moves us closer to isolating k.
Subtracting 3 from both sides, we get:
3 + 4k - 3 = 5b - 3
This simplifies to:
4k = 5b - 3
Now, the term with k is isolated on the left side, making the next step, solving for k, more straightforward. By isolating the variable term, we focus our attention on the final operation needed to find the value of k. This step is essential in the process of solving linear equations, as it sets up the final calculation.
Step 3: Solve for k
Now that we have isolated the term with k, we can solve for k itself. Our equation currently looks like this:
4k = 5b - 3
To find the value of k, we need to get k by itself. This can be achieved by dividing both sides of the equation by the coefficient of k, which is 4. Dividing both sides by 4 ensures that the equation remains balanced and allows us to isolate k.
Dividing both sides by 4, we get:
(4k) / 4 = (5b - 3) / 4
This simplifies to:
k = (5b - 3) / 4
This is the solution for k in terms of b. The value of k is expressed as a function of b, meaning that for any value of b, we can determine the corresponding value of k. By performing this final step, we have successfully solved the equation for k, expressing it in its simplest form. This solution provides a clear and direct relationship between k and b, demonstrating the power of algebraic manipulation in solving linear equations.
Final Solution
After following the step-by-step process, we have successfully solved the equation 3 - 3k + 7k = 5b for k. The final solution is:
k = (5b - 3) / 4
This solution expresses k in terms of b, which means that the value of k depends on the value of b. For any given value of b, we can substitute it into this equation to find the corresponding value of k. This is a common outcome when solving equations with multiple variables; the solution often represents a relationship between the variables rather than a single numerical value.
Conclusion
In this article, we have thoroughly demonstrated how to solve the equation 3 - 3k + 7k = 5b for k. By following a clear, step-by-step approach, we combined like terms, isolated the term with k, and ultimately solved for k. The final solution, k = (5b - 3) / 4, provides a general expression for k in terms of b. This process highlights the importance of algebraic manipulation in solving linear equations. Each step we took was crucial in simplifying the equation and isolating the variable of interest. Mastering these techniques is essential for anyone studying algebra or related fields. By understanding the principles behind each step, you can apply these methods to solve a wide range of algebraic problems. This article not only provides a solution to a specific equation but also reinforces the fundamental skills needed to tackle more complex algebraic challenges. Remember, practice is key to mastering these concepts, so try applying these steps to similar equations to further solidify your understanding. With a solid grasp of these techniques, you'll be well-equipped to handle various algebraic problems with confidence.
