Solving Linear Equations A Step-by-Step Guide To 2x - 13 = 6x + 23

Solving linear equations is a fundamental skill in algebra and forms the basis for more advanced mathematical concepts. This article provides a comprehensive guide on how to solve the linear equation $2x-13=6x+23$ 2x - 13 = 6x + 23 }, explaining each step in detail and offering insights into the underlying principles. Whether you are a student learning algebra for the first time or someone looking to refresh your skills, this guide will help you master the art of isolating variables and finding solutions. Linear equations, characterized by their straight-line graphs, are ubiquitous in various fields, from physics and engineering to economics and computer science. Understanding how to solve them is not just an academic exercise; it’s a crucial tool for problem-solving in the real world. This article aims to equip you with the knowledge and confidence to tackle any linear equation that comes your way. We will begin by breaking down the given equation, ${ 2x - 13 = 6x + 23$, and then meticulously walk through each step required to isolate the variable ${ x }$. Along the way, we will highlight key concepts and techniques, ensuring that you grasp the logic behind each operation. Our goal is not just to provide the answer but to foster a deeper understanding of the algebraic principles involved. So, let’s embark on this journey together and unlock the secrets of solving linear equations.

Step-by-Step Solution

1. Understand the Equation

The equation we aim to solve is ${ 2x - 13 = 6x + 23 }$. Our primary goal is to isolate the variable ${ x }$ on one side of the equation. This involves rearranging the terms so that all terms containing ${ x }$ are on one side and all constant terms are on the other. The key principle we’ll use throughout this process is the idea of maintaining balance. Whatever operation we perform on one side of the equation, we must also perform on the other side to ensure that the equality holds true. This principle is rooted in the fundamental properties of equality, which dictate that adding, subtracting, multiplying, or dividing both sides of an equation by the same value does not change the solution. Before diving into the algebraic manipulations, it’s crucial to have a clear understanding of the equation’s structure. We have two expressions separated by an equals sign, each containing a term with ${ x }$ and a constant term. The challenge lies in strategically maneuvering these terms to achieve our goal of isolating ${ x }$. This requires careful attention to detail and a systematic approach. In the following steps, we will demonstrate how to effectively rearrange the terms using addition, subtraction, and potentially other operations to achieve the desired isolation. Remember, the ultimate aim is to simplify the equation until we have ${ x }$ by itself on one side, revealing its value.

2. Move the x Terms to One Side

To begin isolating ${ x }$, we need to gather all terms containing ${ x }$ on one side of the equation. A common strategy is to move the term with the smaller coefficient of ${ x }$ to the side with the larger coefficient. In our equation, ${ 2x - 13 = 6x + 23 }$, the smaller coefficient is 2. To move the ${ 2x }$ term from the left side to the right side, we subtract ${ 2x }$ from both sides of the equation. This operation maintains the balance of the equation while effectively transferring the ${ x }$ term. Subtracting ${ 2x }$ from both sides gives us: ${ 2x - 13 - 2x = 6x + 23 - 2x }$. On the left side, ${ 2x }$ and ${ -2x }$ cancel each other out, leaving us with just the constant term ${ -13 }$. On the right side, we combine the ${ x }$ terms: ${ 6x - 2x = 4x }$. So, the equation now simplifies to: ${ -13 = 4x + 23 }$. This step is crucial because it consolidates the ${ x }$ terms into a single expression, making it easier to isolate ${ x }$ in the subsequent steps. By strategically subtracting ${ 2x }$ from both sides, we have taken a significant step towards solving the equation. The next step will focus on moving the constant terms to the other side, further simplifying the equation and bringing us closer to our goal.

3. Move the Constants to the Other Side

Now that we have consolidated the ${ x }$ terms on one side of the equation, our next goal is to move all the constant terms to the other side. In our current equation, ${ -13 = 4x + 23 }$, the constant term ${ 23 }$ is on the same side as the ${ x }$ term. To isolate the ${ x }$ term, we need to eliminate this constant. We can achieve this by subtracting ${ 23 }$ from both sides of the equation. This operation maintains the equation's balance while effectively moving the constant term to the opposite side. Subtracting ${ 23 }$ from both sides gives us: ${ -13 - 23 = 4x + 23 - 23 }$. On the left side, we have ${ -13 - 23 }$, which simplifies to ${ -36 }$. On the right side, ${ 23 }$ and ${ -23 }$ cancel each other out, leaving us with just the term ${ 4x }$. So, the equation now simplifies to: ${ -36 = 4x }$. This step is a crucial milestone in solving the equation. By moving the constant terms to one side, we have further isolated the ${ x }$ term, bringing us closer to finding the value of ${ x }$. The next and final step will involve dividing both sides by the coefficient of ${ x }$ to completely isolate the variable.

4. Isolate x

With the equation now in the simplified form ${ -36 = 4x }$, the final step is to isolate ${ x }$. This involves dividing both sides of the equation by the coefficient of ${ x }$, which in this case is 4. Dividing both sides by 4 ensures that we maintain the balance of the equation while effectively isolating ${ x }$. Dividing both sides by 4 gives us: ${ \frac{-36}{4} = \frac{4x}{4} }$. On the left side, ${ \frac{-36}{4} }$ simplifies to ${ -9 }$. On the right side, the 4 in the numerator and the 4 in the denominator cancel each other out, leaving us with just ${ x }$. Therefore, the equation simplifies to: ${ -9 = x }$. This is the solution to the equation. We have successfully isolated ${ x }$ and found its value to be ${ -9 }$. This step-by-step process demonstrates the power of algebraic manipulation in solving linear equations. By systematically applying the principles of equality, we can rearrange terms and isolate variables to find their values. In the next section, we will verify this solution to ensure its accuracy.

Verification of the Solution

To ensure the accuracy of our solution, it is essential to verify it by substituting the value we found for ${ x }$ back into the original equation. Our solution is ${ x = -9 }$, and the original equation is ${ 2x - 13 = 6x + 23 }$. We will now substitute ${ -9 }$ for ${ x }$ in both sides of the equation and check if the equality holds true. Substituting ${ x = -9 }$ into the left side of the equation, we get: ${ 2(-9) - 13 }$. This simplifies to ${ -18 - 13 }$, which equals ${ -31 }$. So, the left side of the equation evaluates to ${ -31 }$. Next, we substitute ${ x = -9 }$ into the right side of the equation, which gives us: ${ 6(-9) + 23 }$. This simplifies to ${ -54 + 23 }$, which also equals ${ -31 }$. So, the right side of the equation evaluates to ${ -31 }$ as well. Since both the left side and the right side of the equation equal ${ -31 }$ when ${ x = -9 }$, we can confidently conclude that our solution is correct. This verification process is a crucial step in solving equations, as it helps to identify any potential errors made during the algebraic manipulations. By substituting the solution back into the original equation and checking for equality, we can ensure the accuracy of our result. In this case, our verification confirms that ${ x = -9 }$ is indeed the correct solution to the equation ${ 2x - 13 = 6x + 23 }$.

Conclusion

In conclusion, we have successfully solved the equation ${ 2x - 13 = 6x + 23 }$ by systematically isolating the variable ${ x }$. Through a step-by-step process, we first moved the ${ x }$ terms to one side, then moved the constant terms to the other side, and finally, we divided both sides by the coefficient of ${ x }$ to find the solution ${ x = -9 }$. We then verified this solution by substituting it back into the original equation and confirming that both sides of the equation were equal. Mastering the art of solving linear equations is a fundamental skill in mathematics, with applications spanning various fields. The techniques we have demonstrated here, such as maintaining balance by performing the same operations on both sides of the equation, are applicable to a wide range of algebraic problems. By understanding these principles and practicing regularly, you can develop confidence in your ability to solve linear equations and tackle more complex mathematical challenges. The key to success in algebra lies in a clear understanding of the underlying concepts and a methodical approach to problem-solving. Remember to always verify your solutions to ensure accuracy and to build a strong foundation in algebraic principles. As you continue your mathematical journey, the skills you have gained in solving linear equations will serve as a valuable asset in your academic and professional endeavors. Embrace the challenge, practice diligently, and you will unlock the power of algebra to solve a multitude of problems. Linear equations are not just abstract concepts; they are tools that can help us understand and model the world around us. So, keep practicing, keep exploring, and keep solving!