Solving The Equation 8x = 7x + 5 And Verifying The Solution

In the realm of mathematics, solving equations is a fundamental skill. It involves finding the value(s) of the unknown variable(s) that make the equation true. This article will delve into the step-by-step process of solving the linear equation $8x = 7x + 5$ and subsequently verifying the solution to ensure accuracy. Mastering this process is crucial for tackling more complex algebraic problems. The core concept revolves around isolating the variable on one side of the equation by performing the same operations on both sides, maintaining the balance and equality. This systematic approach guarantees that the solution obtained is indeed the correct one. Understanding the underlying principles of equation solving not only enhances your mathematical proficiency but also sharpens your problem-solving abilities in various fields. So, let's embark on this journey of unraveling the equation and discovering the value of 'x'. This exploration will not only solidify your understanding of algebraic manipulations but also instill a sense of confidence in tackling similar challenges in the future. Remember, practice is the key to mastering these skills, and this article serves as a stepping stone towards your mathematical prowess. Furthermore, the ability to solve equations forms the bedrock of numerous scientific and engineering applications, making it an indispensable tool in your academic and professional endeavors. By grasping the nuances of equation solving, you empower yourself to navigate a wide spectrum of real-world problems with greater ease and efficiency. So, let's dive in and unlock the secrets hidden within this equation.

Step 1: Isolate the Variable Term

The primary goal in solving any equation is to isolate the variable on one side. In this case, our equation is $8x = 7x + 5$. To isolate the 'x' term, we need to eliminate the $7x$ term from the right side of the equation. This can be achieved by subtracting $7x$ from both sides of the equation. This step is crucial because it maintains the balance of the equation. Whatever operation is performed on one side must be mirrored on the other side to preserve the equality. Subtracting $7x$ from both sides gives us: $8x - 7x = 7x + 5 - 7x$. This operation simplifies the equation by combining like terms. On the left side, $8x - 7x$ simplifies to $x$, and on the right side, $7x - 7x$ cancels out, leaving us with just the constant term, 5. This simplification process is the heart of equation solving, allowing us to gradually peel away the layers of complexity and reveal the solution. The act of subtracting $7x$ from both sides is not arbitrary; it is a deliberate maneuver designed to bring us closer to the ultimate goal of isolating 'x'. This meticulous approach is what distinguishes mathematical problem-solving from guesswork. By understanding the rationale behind each step, you not only solve the equation but also gain a deeper appreciation for the elegance and precision of mathematics. This foundational understanding will serve you well as you encounter more intricate equations and mathematical challenges in the future. Therefore, let's proceed to the next step, carrying with us the knowledge gained from this crucial manipulation.

Step 2: Simplify the Equation

After subtracting $7x$ from both sides, the equation now reads $8x - 7x = 7x + 5 - 7x$. The next step involves simplifying both sides of the equation by combining like terms. On the left side, we have $8x - 7x$, which simplifies to $1x$ or simply $x$. On the right side, we have $7x - 7x + 5$. The $7x$ and $-7x$ terms cancel each other out, leaving us with just $5$. Therefore, the simplified equation is $x = 5$. This simplification is a pivotal moment in the solving process. It represents the culmination of our efforts to isolate the variable. The equation $x = 5$ directly tells us the value of 'x' that satisfies the original equation. This is the solution we have been seeking. The elegance of this step lies in its clarity and conciseness. It encapsulates the entire process of equation solving into a single, definitive statement. The simplification process is not merely a mechanical step; it is an act of revealing the hidden truth within the equation. By combining like terms, we are essentially stripping away the extraneous elements and exposing the core relationship between the variable and the constant. This ability to simplify complex expressions is a hallmark of mathematical proficiency. It allows us to transform seemingly intractable problems into manageable ones. The equation $x = 5$ is the endpoint of our journey, but it is also the starting point for the next crucial step: verification. Before we can confidently declare victory, we must ensure that this solution indeed holds true when substituted back into the original equation. So, let's proceed to the verification step with a sense of anticipation and rigor.

Step 3: Check the Solution

To ensure the solution $x = 5$ is correct, we must substitute this value back into the original equation, $8x = 7x + 5$. This process is known as verification, and it is a crucial step in problem-solving. It serves as a safeguard against potential errors made during the simplification process. By substituting the solution back into the original equation, we are essentially testing whether the equality holds true. If the equation remains balanced after the substitution, then we can confidently assert that our solution is correct. Conversely, if the equation becomes unbalanced, it indicates that an error has been made, and we need to revisit the steps. In our case, substituting $x = 5$ into the left side of the equation, $8x$, gives us $8 * 5 = 40$. Now, substituting $x = 5$ into the right side of the equation, $7x + 5$, gives us $(7 * 5) + 5 = 35 + 5 = 40$. We observe that both sides of the equation evaluate to 40 when $x = 5$. This confirms that our solution is indeed correct. The act of verification is not just a formality; it is an integral part of the mathematical process. It instills a sense of confidence and completeness in our solution. Furthermore, it reinforces the importance of accuracy and attention to detail in mathematics. By meticulously checking our work, we not only avoid errors but also deepen our understanding of the underlying concepts. The verification process is a testament to the rigor and precision that characterize mathematical thinking. It is a skill that extends beyond the realm of equations and applies to all areas of problem-solving. So, having successfully verified our solution, we can now confidently conclude that $x = 5$ is the correct answer to the equation $8x = 7x + 5$.

Conclusion

In conclusion, we have successfully solved the equation $8x = 7x + 5$ and verified the solution. The steps involved subtracting $7x$ from both sides, simplifying the equation to $x = 5$, and then substituting this value back into the original equation to confirm its validity. The verification process yielded $40 = 40$, demonstrating that our solution is indeed correct. This exercise highlights the importance of a systematic approach in solving algebraic equations. By following a step-by-step method, we can break down complex problems into manageable steps. The ability to solve equations is a fundamental skill in mathematics, with applications spanning various fields, including science, engineering, and economics. Mastering this skill requires a solid understanding of algebraic principles and a commitment to accuracy. The process of equation solving is not just about finding the answer; it is also about developing critical thinking and problem-solving skills. Each step in the process requires careful consideration and logical reasoning. The act of verification reinforces the importance of accuracy and attention to detail. By checking our work, we ensure that our solution is not only correct but also reflects a thorough understanding of the problem. The journey of solving this equation has taken us through the core principles of algebra, demonstrating the power and elegance of mathematical reasoning. The solution, $x = 5$, is not just a number; it is the culmination of a logical process that has transformed a complex equation into a simple, verifiable statement. This is the essence of mathematics: the ability to uncover the hidden truths that lie beneath the surface of seemingly intricate problems. So, let us carry forward this newfound knowledge and confidence as we tackle more challenging mathematical endeavors in the future.

Therefore, the solution to the equation $8x = 7x + 5$ is $x = 5$. This detailed walkthrough demonstrates the methodical approach required for solving linear equations and emphasizes the significance of verifying the solution. Remember, consistent practice is essential for mastering these skills.