Solving The Equation X / 35 = 7 A Step By Step Guide
In this article, we delve into the fundamental algebraic equation x / 35 = 7, aiming to provide a comprehensive and easily understandable solution for finding the value of x. This type of equation, a simple yet crucial aspect of algebra, forms the building blocks for more complex mathematical problems. Our focus will be on breaking down the equation, explaining the steps involved in isolating x, and ultimately arriving at the correct answer. We will explore the underlying principles of algebraic manipulation, emphasizing the importance of maintaining balance on both sides of the equation. This approach not only helps in solving the specific problem at hand but also equips readers with the tools to tackle similar equations with confidence. Understanding these basic algebraic concepts is essential for anyone venturing into mathematics, as they appear in various forms across different mathematical disciplines. Whether you're a student just starting with algebra or someone looking to refresh your understanding, this guide will offer a clear and concise explanation of how to solve equations like x / 35 = 7. This foundational knowledge is key to unlocking more advanced mathematical concepts and problem-solving techniques. So, let's embark on this mathematical journey together and unravel the mystery behind solving for x.
Understanding the Equation: x / 35 = 7
At its core, the equation x / 35 = 7 represents a simple division problem. The variable x stands for an unknown number that, when divided by 35, yields the result 7. To solve for x, our goal is to isolate it on one side of the equation. This means we need to perform an operation that will undo the division by 35. The key principle in solving algebraic equations is to maintain balance. Whatever operation we perform on one side of the equation, we must perform the same operation on the other side to ensure the equality remains valid. In this case, the inverse operation of division is multiplication. Therefore, to eliminate the division by 35, we will multiply both sides of the equation by 35. This step is crucial as it allows us to isolate x and determine its value. The equation x / 35 = 7 is a straightforward example of a linear equation, where the variable x is raised to the power of 1. Solving such equations is a fundamental skill in algebra, and mastering it opens the door to tackling more complex mathematical problems. Understanding the relationship between division and multiplication is essential in this process, as it allows us to manipulate the equation effectively and arrive at the correct solution. Let's now delve into the step-by-step process of solving for x.
Step-by-Step Solution: Isolating x
The key to solving the equation x / 35 = 7 lies in isolating the variable x. As we discussed earlier, this involves performing the inverse operation of division, which is multiplication. To begin, we multiply both sides of the equation by 35. This can be written as: (x / 35) * 35 = 7 * 35. On the left side of the equation, multiplying x / 35 by 35 effectively cancels out the division, leaving us with just x. This is because 35 divided by 35 equals 1, and x multiplied by 1 is simply x. On the right side of the equation, we have 7 multiplied by 35. Performing this multiplication gives us 245. Therefore, the equation now becomes x = 245. This result tells us that the value of x that satisfies the original equation is 245. To verify our solution, we can substitute 245 back into the original equation: 245 / 35 = 7. Indeed, 245 divided by 35 equals 7, confirming that our solution is correct. This step-by-step process demonstrates the power of algebraic manipulation in solving equations. By applying the principle of inverse operations and maintaining balance on both sides of the equation, we can effectively isolate the variable and find its value. Understanding this process is crucial for tackling more complex algebraic problems.
Final Answer: x = 245
Having walked through the step-by-step solution, we have confidently arrived at the final answer: x = 245. This means that the unknown number, represented by x, which when divided by 35 gives 7, is 245. This result is not just a numerical value; it represents the solution to the equation x / 35 = 7. To reiterate, we reached this conclusion by applying the fundamental principle of algebraic manipulation: performing the same operation on both sides of the equation to maintain balance and isolate the variable. In this case, we multiplied both sides of the equation by 35, effectively undoing the division and revealing the value of x. The final answer, x = 245, is a testament to the power of these basic algebraic techniques. It underscores the importance of understanding inverse operations and the concept of equality in mathematical problem-solving. This solution not only answers the specific question posed but also reinforces the broader principles of algebra, which are applicable across a wide range of mathematical contexts. So, with x = 245 in hand, we have successfully navigated this algebraic challenge and strengthened our understanding of equation-solving techniques.
Why A. 245 is the Correct Answer
Option A, 245, is the correct answer because it is the only value that satisfies the equation x / 35 = 7. As we demonstrated in the step-by-step solution, multiplying both sides of the equation by 35 isolates x and reveals its value to be 245. This is a direct application of the principles of algebra, where we manipulate equations while maintaining equality. The other options, B. 35, C. 7, and D. 5, are incorrect because they do not satisfy the original equation. If we substitute 35 for x, we get 35 / 35 = 1, which is not equal to 7. Similarly, substituting 7 for x gives us 7 / 35 = 0.2, and substituting 5 for x gives us 5 / 35 ≈ 0.143, neither of which equals 7. Only when x is 245 does the equation hold true. This highlights the importance of not only understanding the steps involved in solving an equation but also verifying the solution by substituting it back into the original equation. This verification step ensures that the answer is correct and that no errors were made during the solving process. Therefore, the correctness of option A, 245, is firmly established through both the algebraic solution and the verification process. This reinforces the importance of a systematic approach to problem-solving in mathematics.
Common Mistakes and How to Avoid Them
When solving equations like x / 35 = 7, several common mistakes can lead to incorrect answers. One frequent error is performing the wrong operation to isolate x. For instance, instead of multiplying both sides by 35, some might mistakenly divide by 35. This would lead to an incorrect result because it exacerbates the division rather than canceling it out. Another common mistake is failing to maintain balance in the equation. This occurs when an operation is performed on one side of the equation but not the other. For example, if someone multiplies the left side by 35 but forgets to do the same on the right side, the equality is broken, and the solution will be incorrect. A third mistake is miscalculating the arithmetic. Simple errors in multiplication or division can lead to a wrong answer, even if the correct steps are followed. For instance, if 7 multiplied by 35 is incorrectly calculated, the final value of x will be wrong. To avoid these mistakes, it's crucial to follow a systematic approach. First, clearly identify the operation needed to isolate x. In this case, it's multiplication. Second, remember to perform the same operation on both sides of the equation. Third, double-check all arithmetic calculations to ensure accuracy. Finally, verify your solution by substituting it back into the original equation. This will help catch any errors and ensure that the answer is correct. By being mindful of these common pitfalls and following a careful, methodical approach, you can significantly improve your accuracy in solving algebraic equations.
Conclusion: Mastering Algebraic Equations
In conclusion, solving the equation x / 35 = 7 provides a valuable lesson in the fundamentals of algebra. Through a step-by-step approach, we've demonstrated how to isolate the variable x by applying the inverse operation of multiplication, ultimately arriving at the solution x = 245. This process underscores the importance of maintaining balance in equations and performing the same operations on both sides to preserve equality. We've also highlighted common mistakes that can occur during equation-solving, such as performing the wrong operation, failing to maintain balance, and making arithmetic errors. By being aware of these pitfalls and adopting a systematic approach, you can minimize the risk of errors and improve your accuracy. Mastering algebraic equations like this one is a crucial step in building a strong foundation in mathematics. These skills are not only essential for academic success but also for problem-solving in various real-world contexts. The ability to manipulate equations, isolate variables, and arrive at correct solutions is a powerful tool that can be applied across many disciplines. Therefore, understanding the principles and techniques discussed in this article will serve you well in your mathematical journey and beyond. Keep practicing, stay mindful of the fundamentals, and you'll find that algebra, once daunting, becomes an accessible and rewarding field of study.
