Solving 6k + 10.5 = 3k + 12 A Step-by-Step Guide To Linear Equations
What is the solution to the linear equation 6k + 10.5 = 3k + 12?
In the realm of mathematics, linear equations serve as foundational tools for modeling various real-world scenarios. Understanding how to solve these equations is crucial for success in algebra and beyond. This article will delve into the process of solving a specific linear equation, providing a step-by-step guide that demystifies the concepts involved. We will explore the equation 6k + 10.5 = 3k + 12, breaking down each step to arrive at the correct solution. By understanding the underlying principles and techniques, you'll be well-equipped to tackle a wide range of linear equations.
Understanding Linear Equations
Before diving into the solution, let's establish a firm understanding of what constitutes a linear equation. A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable. These equations, when graphed, produce a straight line, hence the name "linear." The general form of a linear equation in one variable is often represented as ax + b = c, where x is the variable, and a, b, and c are constants. Solving a linear equation involves isolating the variable on one side of the equation to determine its value. This is achieved by performing the same operations on both sides of the equation, maintaining the balance and equality.
Linear equations are ubiquitous in various fields, from physics and engineering to economics and computer science. They are used to model relationships between quantities, predict outcomes, and solve problems involving unknown values. Mastering the art of solving linear equations is therefore an essential skill for anyone pursuing studies or careers in these areas. The principles and techniques learned in solving linear equations also lay the foundation for understanding more complex mathematical concepts.
The Given Equation: 6k + 10.5 = 3k + 12
Our focus is on the linear equation 6k + 10.5 = 3k + 12. This equation involves the variable k and several constant terms. The goal is to isolate k on one side of the equation to find its numerical value. To achieve this, we will employ a series of algebraic manipulations, ensuring that each step maintains the equation's balance.
This particular equation presents a common scenario in linear equation solving, where the variable appears on both sides of the equation. The initial steps will involve rearranging the terms to group the variable terms on one side and the constant terms on the other. This will pave the way for isolating the variable and determining its value. We will explore the strategic application of inverse operations, a fundamental technique in solving algebraic equations.
Step 1: Grouping the 'k' Terms
The first step in solving the equation 6k + 10.5 = 3k + 12 is to group the terms containing the variable k on one side of the equation. To do this, we can subtract 3k from both sides of the equation. This operation maintains the equality because we are performing the same action on both sides. Subtracting 3k from both sides yields:
6k + 10.5 - 3k = 3k + 12 - 3k
Simplifying this, we combine the k terms on the left side:
(6k - 3k) + 10.5 = 12
This results in:
3k + 10.5 = 12
Now, all the terms with the variable k are on the left side of the equation, setting the stage for the next step in isolating k.
This step highlights the importance of maintaining balance in algebraic manipulations. By subtracting the same term from both sides, we ensure that the equation remains valid and the solution remains unchanged. This principle is fundamental to solving all types of algebraic equations.
Step 2: Isolating the Constant Terms
Having grouped the k terms, the next step is to isolate the constant terms on the other side of the equation. We have the equation 3k + 10.5 = 12. To isolate the constant terms, we need to eliminate the 10.5 from the left side. We can achieve this by subtracting 10.5 from both sides of the equation:
3k + 10.5 - 10.5 = 12 - 10.5
Simplifying, we get:
3k = 1.5
Now, we have the term with k isolated on the left side, and a single constant term on the right side. This brings us closer to finding the value of k.
This step demonstrates the strategic use of inverse operations to isolate variables. By subtracting 10.5, the inverse operation of addition, we effectively removed the constant term from the left side, paving the way for the final step in solving for k.
Step 3: Solving for 'k'
We've arrived at the equation 3k = 1.5. To solve for k, we need to isolate k completely. Currently, k is being multiplied by 3. To undo this multiplication, we perform the inverse operation, which is division. We divide both sides of the equation by 3:
(3k) / 3 = 1.5 / 3
This simplifies to:
k = 0.5
Therefore, the solution to the linear equation 6k + 10.5 = 3k + 12 is k = 0.5.
This final step showcases the culmination of the previous steps, where we strategically applied inverse operations to isolate the variable and determine its value. The result, k = 0.5, represents the value that satisfies the original equation, making the left-hand side equal to the right-hand side.
Verification of the Solution
To ensure the accuracy of our solution, it's always a good practice to verify it by substituting the calculated value of k back into the original equation. Let's substitute k = 0.5 into the equation 6k + 10.5 = 3k + 12:
6(0.5) + 10.5 = 3(0.5) + 12
Simplifying both sides:
3 + 10.5 = 1.5 + 12
13.5 = 13.5
Since both sides of the equation are equal, our solution k = 0.5 is verified as correct.
This verification step underscores the importance of checking your work, especially in mathematics. By substituting the solution back into the original equation, we can confidently confirm its accuracy and avoid potential errors.
Conclusion
In this article, we have thoroughly explored the process of solving the linear equation 6k + 10.5 = 3k + 12. By systematically grouping terms, isolating the variable, and applying inverse operations, we arrived at the solution k = 0.5. We also emphasized the importance of verifying the solution to ensure accuracy. The steps outlined here provide a solid foundation for solving a wide range of linear equations. Mastering these techniques is crucial for success in algebra and related fields. Remember to practice these steps with various equations to solidify your understanding and build confidence in your problem-solving abilities. Linear equations are a fundamental concept in mathematics, and a strong grasp of their solutions will undoubtedly benefit you in your academic and professional pursuits.
The ability to solve linear equations is a cornerstone of mathematical literacy. By understanding the principles and techniques involved, you can confidently tackle a variety of problems and apply these skills to real-world scenarios. This step-by-step guide has provided you with the tools and knowledge necessary to approach linear equations with clarity and precision. Keep practicing, and you'll become a proficient solver of linear equations!
The core concept in solving linear equations is maintaining balance. Every operation performed on one side of the equation must be mirrored on the other side. This ensures that the equality remains intact and the solution remains valid. Understanding this fundamental principle is key to success in solving not just linear equations, but all types of algebraic equations.