Solving X - 2/3 = 5/6 A Step-by-Step Guide
In the realm of mathematics, solving linear equations is a fundamental skill that forms the bedrock of more advanced concepts. Linear equations, characterized by a variable raised to the power of one, are ubiquitous in various fields, from science and engineering to economics and finance. Mastering the techniques to solve these equations is crucial for anyone seeking to delve deeper into the world of mathematics and its applications. Among the various strategies employed, the one-step method stands out for its simplicity and efficiency in tackling specific types of linear equations. This method hinges on the principle of isolating the variable on one side of the equation by performing a single operation on both sides. This operation, carefully chosen, effectively cancels out the term that is preventing the variable from standing alone, thereby revealing its value. Before we dive into the specific equation at hand, it's essential to grasp the core concepts that underpin the one-step method. At its heart lies the golden rule of algebra: whatever operation you perform on one side of the equation, you must perform the same operation on the other side to maintain the balance. This principle ensures that the equality remains valid throughout the solving process. The operations that we can employ to manipulate equations are the basic arithmetic operations: addition, subtraction, multiplication, and division. The choice of which operation to use depends entirely on the structure of the equation we are trying to solve. For instance, if a term is being added to the variable, we would subtract that term from both sides. Conversely, if a term is being subtracted, we would add it to both sides. Similarly, if the variable is being multiplied by a number, we would divide both sides by that number, and if the variable is being divided, we would multiply both sides. The key to success in solving linear equations in one step lies in identifying the operation that will most directly isolate the variable. This requires a keen understanding of the equation's structure and the relationships between its terms. With practice, this skill becomes second nature, allowing you to efficiently solve a wide range of linear equations. Now that we have laid the groundwork, let's turn our attention to the specific equation that we aim to solve: x - 2/3 = 5/6. This equation presents a classic scenario where the one-step method can be applied elegantly to find the value of x.
Understanding the Equation x - 2/3 = 5/6
Before diving into the solution, it's crucial to understand the structure of the equation x - 2/3 = 5/6. This equation is a linear equation because the variable 'x' is raised to the power of one. It states that when 2/3 is subtracted from an unknown value 'x', the result is equal to 5/6. Our goal is to find the value of 'x' that makes this statement true. The equation consists of two sides: the left-hand side (LHS) and the right-hand side (RHS). The LHS is 'x - 2/3', and the RHS is '5/6'. The equals sign (=) signifies that the expressions on both sides have the same value. To solve for 'x', we need to isolate it on one side of the equation. This means we want to manipulate the equation so that 'x' is by itself on either the LHS or the RHS, with its value clearly stated on the other side. In this particular equation, 'x' is being subtracted by 2/3. To isolate 'x', we need to perform the opposite operation, which is addition. We need to add 2/3 to both sides of the equation. This will cancel out the -2/3 on the LHS, leaving 'x' by itself. It's important to remember the golden rule of algebra: whatever operation we perform on one side of the equation, we must perform the same operation on the other side. This ensures that the equation remains balanced and the equality holds true. By adding 2/3 to both sides, we maintain this balance and move closer to isolating 'x'. Understanding this fundamental principle is key to successfully solving linear equations. Now that we have a clear understanding of the equation and the operation required to isolate 'x', we can proceed to the next step: performing the addition on both sides. This will lead us to the solution and reveal the value of 'x'. The ability to deconstruct an equation and identify the necessary steps to solve it is a hallmark of mathematical proficiency. It requires careful observation, logical reasoning, and a solid grasp of algebraic principles. With practice, you can develop this skill and confidently tackle a wide range of equations. In the next section, we will demonstrate the step-by-step process of adding 2/3 to both sides of the equation, showing how this leads to the isolation of 'x' and the determination of its value. This will provide a concrete example of the one-step method in action, solidifying your understanding of this essential problem-solving technique.
The One-Step Solution: Adding 2/3 to Both Sides
To solve the equation x - 2/3 = 5/6 for x in one step, we need to isolate x on one side of the equation. As we discussed earlier, since 2/3 is being subtracted from x, we perform the inverse operation: adding 2/3. However, we must adhere to the golden rule of algebra, which dictates that any operation performed on one side of the equation must also be performed on the other side to maintain equality. Therefore, we add 2/3 to both sides of the equation: x - 2/3 + 2/3 = 5/6 + 2/3. On the left-hand side (LHS), the -2/3 and +2/3 terms cancel each other out, leaving us with just x: x = 5/6 + 2/3. Now, we need to add the fractions on the right-hand side (RHS). To add fractions, they must have a common denominator. The least common denominator (LCD) of 6 and 3 is 6. So, we need to convert 2/3 to an equivalent fraction with a denominator of 6. To do this, we multiply both the numerator and denominator of 2/3 by 2: 2/3 * (2/2) = 4/6. Now we can rewrite the equation as: x = 5/6 + 4/6. Adding the fractions on the RHS, we get: x = (5 + 4) / 6. Simplifying the numerator, we have: x = 9/6. Finally, we can simplify the fraction 9/6 by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 3: x = (9 ÷ 3) / (6 ÷ 3). This gives us the final solution: x = 3/2. Therefore, by adding 2/3 to both sides of the equation in a single step, we have successfully isolated x and found its value to be 3/2. This demonstrates the power and efficiency of the one-step method in solving linear equations. The ability to manipulate equations by performing the same operation on both sides is a cornerstone of algebraic problem-solving. It allows us to transform complex equations into simpler forms, ultimately leading to the isolation of the variable and the determination of its value. In this case, adding 2/3 to both sides effectively undid the subtraction of 2/3 from x, allowing us to directly calculate the value of x. This one-step solution highlights the elegance and conciseness of algebraic methods. By understanding the underlying principles and applying them strategically, we can solve equations efficiently and accurately. In the following sections, we will discuss why the other options provided are incorrect and reinforce the importance of choosing the correct operation to isolate the variable.
Why the Other Options Are Incorrect
Understanding why the correct option is correct is just as important as understanding why the other options are incorrect. This deepens our understanding of the problem-solving process and reinforces the underlying principles of algebra. Let's examine each of the incorrect options provided and explain why they would not lead to the solution of the equation x - 2/3 = 5/6.
Option A: Subtract 2/3 from each side
This option suggests subtracting 2/3 from both sides of the equation. If we were to do this, the equation would become: x - 2/3 - 2/3 = 5/6 - 2/3. On the left-hand side, we would have x minus 2/3 twice, which would result in x - 4/3. This does not isolate x; instead, it further complicates the equation. Subtracting 2/3 from both sides moves us further away from the solution, rather than closer. The goal is to undo the operation that is being performed on x, and subtracting 2/3 again does not achieve this.
Option B: Subtract 5/6 from each side
Subtracting 5/6 from both sides of the equation would result in: x - 2/3 - 5/6 = 5/6 - 5/6. This simplifies to x - 2/3 - 5/6 = 0. While this operation maintains the equality of the equation, it does not isolate x. We still have the -2/3 term on the left-hand side, preventing x from standing alone. Subtracting 5/6 from both sides merely shifts the constant term to the left-hand side, without bringing us any closer to the solution. To isolate x, we need to eliminate the -2/3 term, and subtracting 5/6 does not accomplish this.
Option C: Add 5/6 to each side
Adding 5/6 to both sides of the equation would give us: x - 2/3 + 5/6 = 5/6 + 5/6. This simplifies to x - 2/3 + 5/6 = 10/6. While adding 5/6 to both sides is a valid algebraic manipulation, it does not directly isolate x. We still have the -2/3 term on the left-hand side, which needs to be eliminated. Adding 5/6 to both sides introduces a new constant term on the left-hand side, making the equation more complex rather than simpler. To solve for x in one step, we need to perform the operation that directly cancels out the -2/3 term, and adding 5/6 does not achieve this.
In summary, options A, B, and C are incorrect because they do not effectively isolate x. They either further complicate the equation or simply shift terms around without bringing us closer to the solution. The correct option, option D, involves adding 2/3 to both sides, which is the inverse operation of subtracting 2/3. This cancels out the -2/3 term on the left-hand side, directly isolating x and allowing us to solve for its value. Understanding why these options are incorrect reinforces the importance of choosing the correct operation to isolate the variable when solving linear equations. It highlights the need to carefully analyze the equation and identify the operation that will most directly lead to the solution. By recognizing the effects of different operations on the equation, we can develop a more strategic approach to problem-solving and avoid common pitfalls.
Conclusion: The Power of One-Step Equations
In conclusion, the equation x - 2/3 = 5/6 can be solved for x in one step by adding 2/3 to each side (Option D). This method effectively isolates x, allowing us to determine its value as 3/2. The one-step method is a powerful tool for solving linear equations, but it's crucial to understand the underlying principles and apply them correctly. The golden rule of algebra dictates that any operation performed on one side of the equation must also be performed on the other side to maintain equality. This principle is fundamental to solving equations and ensures that the solution remains valid. The key to success in solving one-step equations lies in identifying the operation that will most directly isolate the variable. This requires careful analysis of the equation and an understanding of inverse operations. In the case of x - 2/3 = 5/6, the variable x is being subtracted by 2/3, so the inverse operation is addition. Adding 2/3 to both sides cancels out the -2/3 term on the left-hand side, leaving x by itself. Understanding why the other options are incorrect is equally important. Subtracting 2/3 from each side, subtracting 5/6 from each side, or adding 5/6 to each side would not effectively isolate x and would either complicate the equation or simply shift terms around without bringing us closer to the solution. The one-step method is not only efficient but also elegant in its simplicity. It demonstrates the power of algebraic manipulation in solving mathematical problems. By mastering this technique, you can build a strong foundation for tackling more complex equations and mathematical concepts. The ability to solve equations is a fundamental skill that is applicable in various fields, from science and engineering to economics and finance. It's a skill that empowers you to analyze problems, make predictions, and solve real-world challenges. So, embrace the power of one-step equations and continue to hone your problem-solving skills. With practice and dedication, you can become proficient in solving equations and unlock the doors to a deeper understanding of mathematics and its applications. The journey of mathematical discovery is filled with challenges and rewards, and mastering the one-step method is a significant milestone along the way. Keep exploring, keep learning, and keep solving!
