Solving 9.9 = 0.2x A Step-by-Step Guide
Introduction
In the realm of mathematics, solving equations is a fundamental skill. It involves isolating a variable to determine its value. This article delves into the process of solving a linear equation for the variable x. We will specifically address the equation 9.9 = 0.2x, providing a step-by-step solution and elucidating the underlying mathematical principles. Our focus will be on ensuring clarity and understanding, making this guide accessible to learners of all levels. To effectively solve equations, a solid grasp of basic algebraic principles is necessary. This includes the ability to perform arithmetic operations (addition, subtraction, multiplication, and division) and an understanding of the properties of equality. The primary goal is to manipulate the equation in such a way that the variable x is isolated on one side, thereby revealing its value. Each step taken must maintain the equality of the equation, which means performing the same operation on both sides. Linear equations, such as the one we are addressing, are characterized by having a variable raised to the power of one. These equations are particularly common in various fields, including science, engineering, and economics. Their straightforward nature allows for easy manipulation and solution, making them a cornerstone of mathematical problem-solving. This discussion will not only provide the solution to the specific equation but also reinforce the general methodology for solving linear equations, empowering readers to tackle similar problems with confidence. The process involves a series of logical steps, each building upon the previous one, to arrive at the final answer. By understanding these steps and the reasoning behind them, readers can develop a deeper appreciation for the power and elegance of mathematical problem-solving. Let's embark on this journey of solving for x and unlock the secrets hidden within the equation 9.9 = 0.2x.
Step-by-Step Solution
The given equation is 9.9 = 0.2x. Our primary goal is to isolate the variable x on one side of the equation. To achieve this, we need to undo the multiplication by 0.2 that is currently associated with x. The fundamental principle we employ here is that we can perform the same operation on both sides of the equation without changing its balance. This ensures that the equality remains valid throughout the solution process. The opposite operation of multiplication is division. Therefore, to isolate x, we will divide both sides of the equation by 0.2. This step is crucial as it directly leads us closer to finding the value of x. Dividing both sides by 0.2 gives us: 9.9 / 0.2 = (0.2x) / 0.2. On the right side of the equation, 0.2 in the numerator and denominator cancel each other out, leaving us with just x. This is exactly what we aimed for – isolating the variable. Now, we need to perform the division on the left side of the equation. Dividing 9.9 by 0.2 can be done manually or with the aid of a calculator. The result of this division is 49.5. Therefore, we have 49.5 = x. This equation tells us that the value of x that satisfies the original equation is 49.5. We have successfully isolated x and found its value. It's always a good practice to verify the solution by substituting it back into the original equation. If the equation holds true after the substitution, we can be confident that our solution is correct. In this case, substituting 49.5 for x in the original equation (9.9 = 0.2x) yields 9.9 = 0.2 * 49.5, which simplifies to 9.9 = 9.9. This confirms that our solution, x = 49.5, is indeed correct. The step-by-step approach ensures clarity and accuracy in solving the equation.
Detailed Calculation
To delve deeper into the solution, let's break down the division of 9.9 by 0.2. This calculation is a crucial step in isolating x and finding its value. When dividing decimals, it's often helpful to eliminate the decimal point in the divisor. We can achieve this by multiplying both the dividend (9.9) and the divisor (0.2) by 10. This maintains the ratio and simplifies the division process. Multiplying 9.9 by 10 gives us 99, and multiplying 0.2 by 10 gives us 2. So, the division problem now becomes 99 ÷ 2. This transformation makes the division easier to perform, either manually or with a calculator. Now, let's perform the division of 99 by 2. 2 goes into 9 four times (2 * 4 = 8), leaving a remainder of 1. We bring down the next digit, which is 9, making the new dividend 19. 2 goes into 19 nine times (2 * 9 = 18), leaving a remainder of 1. Since there are no more whole number digits to bring down, we add a decimal point and a zero to the dividend, making it 99.0. We bring down the 0, making the new dividend 10. 2 goes into 10 five times (2 * 5 = 10), leaving no remainder. Therefore, 99 ÷ 2 = 49.5. This detailed calculation confirms the result we obtained in the previous section. The value of 9.9 / 0.2 is indeed 49.5. This meticulous approach to the division process underscores the importance of accuracy in mathematical calculations. Every step, no matter how small, contributes to the final result. By understanding the mechanics of decimal division, we can confidently solve similar problems and gain a deeper appreciation for the precision of mathematics. The ability to perform such calculations accurately is a valuable skill in various fields, from science and engineering to finance and everyday life. This detailed explanation serves to solidify the understanding of the solution process and reinforce the importance of careful calculation.
Verification of the Solution
As mentioned earlier, verification is a crucial step in the problem-solving process. It ensures that the solution obtained is correct and satisfies the original equation. To verify our solution, x = 49.5, we substitute this value back into the original equation, 9.9 = 0.2x. This substitution will allow us to check if the equation holds true with the calculated value of x. Replacing x with 49.5 in the equation gives us 9.9 = 0.2 * 49.5. Now, we need to perform the multiplication on the right side of the equation. Multiplying 0.2 by 49.5 can be done manually or with a calculator. 0. 2 * 49.5 = 9.9. So, the equation becomes 9.9 = 9.9. This equation is indeed true. The left side of the equation is equal to the right side, confirming that our solution, x = 49.5, is correct. This verification process provides us with confidence in our answer. It demonstrates that we have correctly applied the principles of algebra and arithmetic to solve the equation. Verification is not just a formality; it's a critical step in ensuring the accuracy of our work. In mathematical problem-solving, it's always wise to double-check our answers, especially in situations where errors can have significant consequences. By verifying the solution, we eliminate the possibility of making a mistake and ensure that our understanding of the problem is sound. This practice of verification is a hallmark of a diligent and thorough problem-solver. It reflects a commitment to accuracy and a desire to understand the underlying principles of mathematics. In this case, the verification process has reinforced our confidence in the solution x = 49.5 and has solidified our understanding of the equation 9.9 = 0.2x. The importance of verification cannot be overstated, as it is a cornerstone of accurate and reliable problem-solving.
Conclusion
In conclusion, we have successfully solved the equation 9.9 = 0.2x for x. By following a step-by-step approach, we isolated the variable x and determined its value to be 49.5. The process involved dividing both sides of the equation by 0.2, a crucial step in undoing the multiplication and revealing the value of x. We also delved into the detailed calculation of 9.9 / 0.2, ensuring a thorough understanding of the arithmetic involved. Furthermore, we emphasized the importance of verifying the solution by substituting it back into the original equation. This verification process confirmed that our solution, x = 49.5, is indeed correct. The equation 9. 9 = 0.2 * 49.5 holds true, validating our result. Solving equations is a fundamental skill in mathematics, and this exercise has demonstrated the principles involved in solving linear equations. The ability to isolate a variable and determine its value is essential in various mathematical contexts and real-world applications. This discussion has not only provided the solution to the specific equation but also reinforced the general methodology for solving linear equations. By understanding the steps and the reasoning behind them, readers can confidently tackle similar problems in the future. The clarity and accuracy with which we approached the solution underscore the importance of meticulousness in mathematical problem-solving. Each step, from the initial division to the final verification, contributes to the overall correctness of the answer. The process of solving equations is a journey of logical deduction and careful calculation. By mastering these skills, we empower ourselves to navigate the complexities of mathematics and apply its principles to solve a wide range of problems. The solution x = 49.5 is not just a numerical answer; it's a testament to the power of mathematical reasoning and the importance of a systematic approach.
