Solving For Y In (2y-1)/-3 = -5 A Step-by-Step Guide

#h1 Solving for y: A Deep Dive into the Equation (2y-1)/-3 = -5

In the realm of mathematics, solving for variables is a fundamental skill. Solving for y in the equation (2y-1)/-3 = -5 requires a systematic approach, leveraging the principles of algebra to isolate the variable. This article provides a comprehensive guide to solving this equation, breaking down each step and explaining the underlying concepts. Our main keyword here is solving for y in a fractional equation. We'll explore the step-by-step process, including clearing the fraction, simplifying the equation, and ultimately isolating 'y' to find its value. This process demonstrates key algebraic techniques applicable to a wide range of mathematical problems. Understanding these techniques is crucial for anyone studying algebra or related fields. The equation (2y-1)/-3 = -5 might seem daunting at first, but by following the correct procedure, it can be solved easily. This problem is an excellent example of how to apply the order of operations in reverse to isolate a variable. Additionally, it highlights the importance of maintaining balance on both sides of the equation to ensure accuracy. Let’s embark on this mathematical journey to demystify the equation and empower you with the ability to solve similar problems confidently. We will begin by addressing the fraction, then simplify the equation by performing necessary operations and finally isolate the y variable, revealing its true value. By the end of this article, you'll not only be able to solve this specific equation but also have a clearer understanding of the broader algebraic principles involved.

Understanding the Fundamentals of Algebraic Equations

Before diving into the solution, let's revisit some core algebraic principles. An algebraic equation is a mathematical statement asserting the equality of two expressions. Solving an equation means finding the value(s) of the variable(s) that make the equation true. This often involves isolating the variable on one side of the equation. To isolate the variable, we perform operations on both sides of the equation while maintaining the equality. Common operations include addition, subtraction, multiplication, and division. The key is to apply inverse operations to undo the operations affecting the variable. For instance, if a variable is being multiplied by a number, we divide both sides of the equation by that number. Similarly, if a number is being added to the variable, we subtract that number from both sides. These basic principles form the foundation for solving more complex equations. Understanding these concepts is essential for successfully manipulating equations and arriving at the correct solution. In the context of our equation, (2y-1)/-3 = -5, we will apply these principles to isolate 'y'. We will start by undoing the division by -3, and then address the terms involving 'y' until we have 'y' by itself on one side of the equation. This methodical approach is crucial for accuracy and understanding the logic behind each step. Remember, each step taken must maintain the balance of the equation, ensuring that both sides remain equal. Mastering these fundamentals is not only crucial for solving equations but also for understanding more advanced mathematical concepts.

Step-by-Step Solution to (2y-1)/-3 = -5

Now, let's tackle the equation (2y-1)/-3 = -5 step by step.

Step 1: Clearing the Fraction

The first step in solving this equation is to eliminate the fraction. We can do this by multiplying both sides of the equation by the denominator, which is -3. This is a crucial step because it simplifies the equation and makes it easier to work with. Multiplying both sides by -3, we get: (-3) * [(2y-1)/-3] = (-3) * (-5). On the left side, the -3 in the numerator and denominator cancel out, leaving us with 2y - 1. On the right side, (-3) * (-5) equals 15. So, our equation now becomes 2y - 1 = 15. This step highlights the importance of performing the same operation on both sides of the equation to maintain balance. By clearing the fraction, we have transformed the equation into a simpler form that is easier to solve. This technique is commonly used in algebra to deal with equations involving fractions. The ability to clear fractions efficiently is a valuable skill that simplifies the subsequent steps in solving for the variable. Remember, the goal is to isolate 'y', and this step brings us closer to that goal by removing the fraction.

Step 2: Isolating the Term with 'y'

Next, we need to isolate the term containing 'y', which is 2y. To do this, we undo the subtraction of 1 by adding 1 to both sides of the equation. Adding 1 to both sides of 2y - 1 = 15, we get: 2y - 1 + 1 = 15 + 1. This simplifies to 2y = 16. By adding 1 to both sides, we have successfully isolated the term with 'y' on the left side of the equation. This step is a classic example of applying inverse operations to simplify an equation. The addition property of equality ensures that adding the same value to both sides maintains the balance of the equation. Isolating the term with 'y' is a critical step towards finding the value of 'y' itself. We are now one step closer to our goal. The process of isolating terms is a fundamental skill in algebra, and it's used extensively in solving various types of equations.

Step 3: Solving for 'y'

Finally, to solve for 'y', we need to undo the multiplication by 2. We do this by dividing both sides of the equation by 2. Dividing both sides of 2y = 16 by 2, we get: (2y)/2 = 16/2. This simplifies to y = 8. Therefore, the solution to the equation (2y-1)/-3 = -5 is y = 8. This final step demonstrates the power of inverse operations in isolating a variable. By dividing both sides by 2, we have successfully solved for 'y'. The value y = 8 is the solution that makes the original equation true. This methodical approach, starting from clearing fractions to isolating the variable, is a standard technique in algebra. Understanding and applying these steps will enable you to solve a wide range of equations. The solution y = 8 can be verified by substituting it back into the original equation to confirm that both sides are equal. This step-by-step process not only provides the answer but also reinforces the underlying principles of algebraic manipulation.

Verification of the Solution

To ensure the accuracy of our solution, it's crucial to verify it. We can do this by substituting y = 8 back into the original equation, (2y-1)/-3 = -5, and checking if both sides of the equation are equal. Substituting y = 8 into the equation, we get: (2(8)-1)/-3 = -5. Simplifying the numerator, we have (16-1)/-3 = -5, which further simplifies to 15/-3 = -5. Now, dividing 15 by -3, we get -5 = -5. Since both sides of the equation are equal, our solution y = 8 is correct. Verification is a critical step in problem-solving as it confirms the accuracy of the solution and helps to avoid errors. It also reinforces the understanding of the equation and the solution process. By substituting the solution back into the original equation, we are essentially reversing the steps we took to solve it. This process provides confidence in the correctness of the answer. In mathematics, always verifying your solution is a good practice.

Common Mistakes to Avoid While Solving Equations

While solving equations, it's essential to avoid common mistakes that can lead to incorrect solutions. One frequent mistake is not applying the same operation to both sides of the equation. Remember, maintaining balance is crucial. Another common error is misapplying the order of operations. Make sure to follow the correct order (PEMDAS/BODMAS) when simplifying expressions. Sign errors are also common, especially when dealing with negative numbers. Pay close attention to the signs while performing operations. Another mistake is incorrectly distributing a number or sign across parentheses. Ensure that you multiply each term inside the parentheses by the number outside. Furthermore, not simplifying expressions before solving can lead to confusion and errors. Always simplify both sides of the equation as much as possible before proceeding. Checking the solution by substituting it back into the original equation can help identify mistakes. By being aware of these common pitfalls, you can improve your accuracy and problem-solving skills. In the context of the equation (2y-1)/-3 = -5, be particularly mindful of the negative signs and the order of operations. A systematic approach, combined with attention to detail, will help you avoid these common errors.

Conclusion: Mastering the Art of Solving Equations

In conclusion, solving for y in the equation (2y-1)/-3 = -5 involves a methodical approach, applying fundamental algebraic principles. We successfully isolated 'y' by clearing the fraction, simplifying the equation, and performing inverse operations. The solution, y = 8, was verified by substituting it back into the original equation. This process demonstrates the importance of understanding and applying algebraic concepts such as the properties of equality and the order of operations. Mastering these skills is essential for success in mathematics and related fields. By avoiding common mistakes and practicing regularly, you can enhance your problem-solving abilities. The ability to solve equations is not only a valuable mathematical skill but also a crucial tool in various real-world applications. From science and engineering to finance and economics, the ability to manipulate and solve equations is indispensable. The step-by-step approach outlined in this article provides a solid foundation for tackling more complex equations. Remember, practice makes perfect, so continue to challenge yourself with different types of equations to solidify your understanding. The journey of learning mathematics is a continuous process, and each solved equation is a step forward. Keep practicing and keep exploring the fascinating world of mathematics!