Solve The Equation 15x = 6x + 27 For X.
Introduction
In the realm of mathematics, solving equations is a fundamental skill. This article will guide you through the process of solving a linear equation, specifically the equation 15x = 6x + 27. We'll break down each step in detail, ensuring you understand the logic and techniques involved. Understanding how to solve linear equations is crucial as it forms the basis for more complex mathematical concepts. Whether you're a student learning algebra or simply someone looking to refresh your math skills, this guide provides a clear and concise approach to solving this type of problem. Linear equations appear in various fields, from physics and engineering to economics and computer science. Mastering the ability to solve them empowers you to tackle real-world problems with confidence. So, let's dive into the steps and demystify the process of solving for x in the given equation.
Understanding the Basics of Linear Equations
Before we jump into the specific equation, let's establish a firm foundation by understanding the core principles of linear equations. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The variable is typically represented by letters like x, y, or z, and the highest power of the variable is 1. This means you won't see terms like x² or x³ in a linear equation. The general form of a linear equation in one variable is ax + b = c, where a, b, and c are constants, and x is the variable we aim to solve for. The goal of solving a linear equation is to isolate the variable on one side of the equation, effectively determining its value. To achieve this, we employ a set of algebraic manipulations that maintain the equation's balance. This balance is crucial; any operation performed on one side of the equation must also be performed on the other side to ensure the equality remains valid. Common operations include adding, subtracting, multiplying, and dividing both sides by the same number. These operations are based on the fundamental properties of equality, which allow us to manipulate equations without changing their solutions. By understanding these basic principles, you'll be well-equipped to tackle a wide range of linear equations. Remember, practice is key! The more you work with linear equations, the more comfortable and confident you'll become in solving them.
Step 1: Isolate the Variable Terms
The first key step in solving the equation 15x = 6x + 27 is to isolate the terms containing the variable x on one side of the equation. In this case, we have 15x on the left side and 6x on the right side. To bring these terms together, we need to eliminate the 6x term from the right side. This can be achieved by subtracting 6x from both sides of the equation. Remember, the golden rule of equation solving is that any operation performed on one side must also be performed on the other side to maintain the balance. Subtracting 6x from both sides gives us: 15x - 6x = 6x + 27 - 6x. Now, we simplify each side of the equation. On the left side, 15x - 6x simplifies to 9x. On the right side, 6x - 6x cancels out, leaving us with just 27. Therefore, our equation now becomes: 9x = 27. This step is crucial because it consolidates the variable terms, bringing us closer to isolating x and finding its value. By isolating the variable terms, we create a simpler equation that is easier to solve. This principle of simplification is a fundamental strategy in algebra and is applicable to many different types of equations. So, by subtracting 6x from both sides, we've successfully taken the first step towards solving for x. This sets the stage for the next step, which will involve isolating x completely.
Step 2: Solve for x by Division
Now that we have the simplified equation 9x = 27, the next step is to isolate x completely. The variable x is currently being multiplied by 9. To undo this multiplication and isolate x, we need to perform the inverse operation, which is division. We'll divide both sides of the equation by 9. This is another application of the fundamental principle of equality: whatever operation we perform on one side, we must also perform on the other. Dividing both sides by 9 gives us: (9x) / 9 = 27 / 9. On the left side, the 9 in the numerator and the 9 in the denominator cancel each other out, leaving us with just x. On the right side, 27 divided by 9 equals 3. Therefore, the equation simplifies to: x = 3. This is our solution! We have successfully isolated x and determined its value. This step demonstrates the power of inverse operations in solving equations. By understanding which operation undoes another (subtraction undoes addition, division undoes multiplication, etc.), we can systematically isolate the variable and find its value. In this case, dividing by 9 effectively 'undoes' the multiplication by 9, allowing us to solve for x. The solution x = 3 means that if we substitute 3 for x in the original equation (15x = 6x + 27), the equation will hold true. This highlights the importance of checking our solution to ensure its accuracy.
Step 3: Verification of the Solution
After finding a solution to an equation, it's always a good practice to verify the solution. This ensures that our calculations are correct and that the value we found for x actually satisfies the original equation. In our case, we found that x = 3 is the solution to the equation 15x = 6x + 27. To verify this, we will substitute x = 3 back into the original equation and check if both sides of the equation are equal. Substituting x = 3 into the left side of the equation (15x) gives us: 15 * 3 = 45. Now, we substitute x = 3 into the right side of the equation (6x + 27): 6 * 3 + 27 = 18 + 27 = 45. We can see that both the left side and the right side of the equation equal 45 when x = 3. This confirms that our solution x = 3 is correct. Verification is a crucial step in problem-solving because it helps prevent errors and builds confidence in your answer. It's a simple yet powerful technique that can save you from making mistakes in more complex problems. By substituting the solution back into the original equation, we are essentially checking if the equation holds true for that particular value of x. If the equation holds true, we can be confident that we have found the correct solution. If not, it indicates that there might be an error in our calculations, and we need to re-examine our steps. So, always remember to verify your solutions, especially in more challenging problems.
Conclusion
In this article, we have walked through the process of solving the linear equation 15x = 6x + 27. We started by understanding the basics of linear equations and the fundamental principles of equality. Then, we followed a step-by-step approach: first, we isolated the variable terms by subtracting 6x from both sides, resulting in the equation 9x = 27. Next, we solved for x by dividing both sides by 9, which gave us the solution x = 3. Finally, we verified our solution by substituting x = 3 back into the original equation and confirmed that both sides were equal. This process highlights the importance of a systematic approach to solving equations. By breaking down the problem into smaller, manageable steps, we can simplify the task and avoid errors. The key to success in algebra, and mathematics in general, lies in understanding the underlying principles and practicing regularly. The more you practice solving different types of equations, the more confident and proficient you will become. Linear equations are a foundational concept in mathematics, and mastering them will pave the way for understanding more advanced topics. Remember to always double-check your work and verify your solutions to ensure accuracy. With consistent practice and a clear understanding of the steps involved, you can confidently solve a wide range of linear equations. So, keep practicing, and you'll be well on your way to mastering algebra!