Solve The Equation (1/9)x = 6. Explain The Steps Clearly.

In the realm of mathematics, solving linear equations is a fundamental skill that forms the bedrock for more advanced concepts. A linear equation is essentially an algebraic equation where each term is either a constant or the product of a constant and a single variable. The beauty of linear equations lies in their simplicity and the straightforward methods available to find their solutions. This article delves into the step-by-step process of solving the linear equation (1/9)x = 6, providing a clear and comprehensive guide for anyone seeking to master this essential mathematical skill. Understanding how to solve such equations not only strengthens your mathematical foundation but also equips you with a valuable tool for tackling real-world problems that can be modeled using linear relationships. From determining the cost of items to calculating distances and speeds, the applications of linear equations are vast and varied. So, whether you are a student grappling with algebra or simply someone looking to refresh your mathematical knowledge, this guide will provide you with the necessary insights and techniques to confidently solve linear equations.

Before diving into the solution, it's crucial to understand the structure of the equation (1/9)x = 6. This equation is a classic example of a linear equation in one variable, where 'x' represents the unknown value we aim to find. The fraction 1/9 is the coefficient of 'x', indicating that 'x' is being multiplied by one-ninth. The equals sign (=) signifies that the expression on the left-hand side (1/9)x is equivalent to the value on the right-hand side, which is 6. In essence, the equation is stating that one-ninth of a certain number 'x' is equal to 6. To solve for 'x', we need to isolate it on one side of the equation. This involves performing operations that maintain the equality while gradually simplifying the equation until 'x' stands alone. The key principle to remember is that any operation performed on one side of the equation must also be performed on the other side to preserve the balance. This ensures that the solution we obtain is valid and accurately reflects the value of 'x' that satisfies the original equation. By carefully applying this principle, we can systematically unravel the equation and reveal the hidden value of 'x'.

The most effective way to solve the equation (1/9)x = 6 is to isolate the variable 'x'. This means getting 'x' by itself on one side of the equation. Since 'x' is being multiplied by 1/9, we can undo this multiplication by performing the inverse operation, which is multiplication by the reciprocal of 1/9. The reciprocal of 1/9 is 9/1, which is simply 9. Therefore, to isolate 'x', we multiply both sides of the equation by 9. This step is crucial because it maintains the balance of the equation while simultaneously eliminating the fraction that is attached to 'x'. When we multiply (1/9)x by 9, the 9 in the numerator cancels out the 9 in the denominator, leaving us with just 'x'. On the other side of the equation, we multiply 6 by 9, which gives us 54. This process effectively transforms the original equation into a much simpler form, where 'x' is now isolated and its value is readily apparent. By carefully applying the principle of inverse operations, we have successfully navigated the equation towards its solution.

To perform the calculation, we multiply both sides of the equation (1/9)x = 6 by 9. This gives us: 9 * (1/9)x = 9 * 6. On the left-hand side, the multiplication of 9 and 1/9 effectively cancels each other out, leaving us with just 'x'. This is because 9 multiplied by 1/9 is equal to 1, and 1 multiplied by 'x' is simply 'x'. On the right-hand side, we perform the multiplication 9 * 6, which results in 54. Therefore, the equation simplifies to x = 54. This straightforward calculation demonstrates the power of inverse operations in isolating the variable and revealing its value. By multiplying both sides of the equation by the reciprocal of the coefficient of 'x', we have successfully eliminated the fraction and determined the value of 'x' that satisfies the original equation. This step is a fundamental technique in solving linear equations and underscores the importance of understanding the relationship between multiplication and division.

After performing the multiplication, we arrive at the solution: x = 54. This means that the value of 'x' that satisfies the equation (1/9)x = 6 is 54. In other words, one-ninth of 54 is equal to 6. To verify this solution, we can substitute 54 back into the original equation and check if it holds true. Substituting 54 for 'x' in the equation (1/9)x = 6 gives us (1/9) * 54 = 6. Simplifying the left-hand side, we find that (1/9) * 54 is indeed equal to 6, confirming that our solution is correct. This verification step is a crucial practice in mathematics, as it ensures that the solution we have obtained is accurate and valid. The solution x = 54 represents the final answer to the problem, and it demonstrates the successful application of algebraic principles to solve a linear equation. By systematically isolating the variable and performing the necessary calculations, we have confidently determined the value of 'x' that satisfies the given equation.

Verification is a crucial step in solving any mathematical equation, as it ensures the accuracy of the solution. To verify that x = 54 is the correct solution for the equation (1/9)x = 6, we substitute 54 back into the original equation. This means replacing 'x' with 54 in the equation (1/9)x = 6, which gives us (1/9) * 54 = 6. Now, we simplify the left-hand side of the equation. Multiplying 1/9 by 54 is the same as dividing 54 by 9, which results in 6. Therefore, the left-hand side simplifies to 6, and the equation becomes 6 = 6. This equality is true, which confirms that our solution x = 54 is indeed correct. The verification process not only provides assurance that the solution is accurate but also reinforces the understanding of the equation and the relationship between the variable and the constants. By systematically substituting the solution back into the original equation, we can confidently validate our answer and avoid potential errors. This practice is a cornerstone of mathematical rigor and should be applied whenever solving equations.

While multiplying both sides by the reciprocal is a standard method for solving this type of equation, another approach is cross-multiplication. To use cross-multiplication, we can rewrite the equation (1/9)x = 6 as x/9 = 6/1. Cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa. In this case, we multiply x by 1 and 9 by 6. This gives us the equation 1 * x = 9 * 6, which simplifies to x = 54. This method provides a slightly different perspective on solving the equation but ultimately arrives at the same solution. Cross-multiplication is a particularly useful technique when dealing with equations that involve fractions on both sides. It allows us to eliminate the fractions and transform the equation into a simpler form that is easier to solve. While it may not be necessary for this specific equation, understanding cross-multiplication broadens our mathematical toolkit and provides an alternative approach for tackling similar problems. This versatility is valuable in mathematics, as it allows us to choose the method that best suits the situation and our individual preferences.

In conclusion, solving the equation (1/9)x = 6 demonstrates the fundamental principles of solving linear equations. By isolating the variable 'x' through multiplication by the reciprocal or by using cross-multiplication, we arrive at the solution x = 54. The verification step further solidifies our understanding and confirms the accuracy of the solution. Mastering linear equations is crucial for success in mathematics, as they form the foundation for more complex algebraic concepts. The ability to confidently solve linear equations not only enhances problem-solving skills but also provides a valuable tool for tackling real-world applications. From calculating financial transactions to determining geometric relationships, linear equations are ubiquitous in various fields of study and everyday life. Therefore, investing time and effort in mastering these fundamental skills will undoubtedly yield significant benefits in the long run. By understanding the principles and techniques outlined in this guide, anyone can confidently approach and solve linear equations, paving the way for further mathematical exploration and achievement.