Solving The Equation -7y = 2.31 A Step-by-Step Guide

In the realm of mathematics, solving linear equations is a fundamental skill. These equations, characterized by a variable raised to the first power, appear in various contexts, from simple algebra problems to complex scientific calculations. This article provides a detailed, step-by-step guide to solving the equation -7y = 2.31, illustrating the core principles involved in isolating the variable and finding its value. Understanding how to manipulate equations and isolate variables is crucial not only for academic success but also for practical problem-solving in everyday life. Before diving into the solution, let's establish a solid foundation by briefly discussing the basics of linear equations and the principles behind solving them. A linear equation, at its heart, is a mathematical statement that asserts the equality between two expressions. These expressions contain one or more variables, but in the simplest form, we deal with equations containing a single variable, typically denoted as 'x' or, in this case, 'y'. The goal in solving a linear equation is to find the value of the variable that makes the equation true. This value is known as the solution to the equation. The key principle we employ to achieve this is maintaining the balance of the equation. Think of an equation as a balanced scale; whatever you do to one side, you must also do to the other to keep the scale balanced. This principle underpins the operations we perform to isolate the variable, such as adding, subtracting, multiplying, or dividing both sides of the equation by the same quantity. With this foundational understanding in place, let's move on to solving the specific equation at hand: -7y = 2.31. We will break down the process into clear, manageable steps, ensuring that the underlying logic is fully transparent. This will not only help you understand the solution to this particular problem but also equip you with the skills to tackle a wide range of linear equations with confidence.

Step-by-Step Solution to -7y = 2.31

When confronted with the equation -7y = 2.31, the primary objective is to isolate the variable 'y' on one side of the equation. This involves systematically undoing the operations that are being performed on 'y'. In this case, 'y' is being multiplied by -7. To isolate 'y', we need to perform the inverse operation, which is division. We will divide both sides of the equation by -7, maintaining the balance of the equation as we discussed earlier. This ensures that the equality remains valid and we are progressing towards the correct solution. The decision to divide by -7 is crucial because it cancels out the -7 that is currently multiplying 'y'. This leaves 'y' standing alone on the left-hand side, which is precisely what we want. The process of dividing both sides by the coefficient of the variable is a fundamental technique in solving linear equations, and it's essential to grasp this concept to solve more complex problems. By applying this principle carefully and accurately, we can systematically unravel the equation and reveal the value of 'y'. This step-by-step approach not only leads us to the solution but also reinforces the importance of maintaining balance in mathematical operations. Let's now proceed with the actual division and see how it unveils the solution to our equation.

Step 1: Divide both sides by -7

To isolate 'y', we divide both sides of the equation by -7:

(-7y) / -7 = 2.31 / -7

This step is crucial because dividing both sides by the same number maintains the equation's balance. When we divide -7y by -7, the -7 in the numerator and the -7 in the denominator cancel each other out, leaving us with just 'y' on the left-hand side. This is precisely the goal we set out to achieve – to isolate the variable. On the right-hand side, we have 2.31 divided by -7, which is a straightforward arithmetic operation. Performing this division will give us the value of 'y'. The importance of this step cannot be overstated. It demonstrates a core principle in algebra: performing the same operation on both sides of an equation preserves the equality. This principle allows us to manipulate equations in a way that progressively simplifies them, bringing us closer to the solution. By carefully applying this principle, we can solve a wide variety of algebraic problems. Now, let's proceed with the division on the right-hand side and determine the numerical value of 'y'. This will complete the process of solving the equation and provide us with the answer we've been seeking.

Step 2: Simplify the equation

Simplifying the equation after dividing both sides by -7, we get:

y = -0.33

This result is obtained by performing the division 2.31 / -7. The negative sign is crucial here; it indicates that the value of 'y' is a negative number. This is a direct consequence of dividing a positive number (2.31) by a negative number (-7). The simplification step is essential because it presents the solution in its most concise and understandable form. After performing the division, we arrive at a single numerical value for 'y', which is the solution to the original equation. This value represents the point where the equation -7y = 2.31 holds true. In other words, if we substitute -0.33 for 'y' in the original equation, the left-hand side will equal the right-hand side. This step also underscores the importance of accuracy in calculations. A small error in the division can lead to a completely different solution. Therefore, it's always a good practice to double-check your calculations, especially when dealing with decimals and negative numbers. Now that we have found the solution, it's a good idea to verify it to ensure its correctness. This will be the focus of the next section.

Verification of the Solution

To ensure the accuracy of our solution, it is vital to substitute the calculated value of 'y' back into the original equation. This process, known as verification, acts as a crucial check, confirming whether our solution truly satisfies the equation. By replacing 'y' with -0.33 in the equation -7y = 2.31, we can directly assess if both sides of the equation are equal. This step is not merely a formality; it's a fundamental practice in mathematics that helps prevent errors and reinforces understanding. Verification is particularly important when dealing with equations involving decimals or negative numbers, as these can sometimes lead to calculation mistakes. Furthermore, the verification process deepens our comprehension of what it means to solve an equation. It highlights the fact that the solution is a value that, when plugged back into the equation, makes the equation a true statement. This concept is essential for building a strong foundation in algebra and beyond. Let's now proceed with the actual substitution and calculation to see if our solution holds up under scrutiny. This will provide us with the confidence that we have indeed found the correct value for 'y'.

Substitute y = -0.33 into the original equation

Substituting y = -0.33 into the original equation -7y = 2.31, we get:

-7 * (-0.33) = 2.31

This substitution replaces the variable 'y' with the numerical value we calculated in the previous steps. It transforms the equation from a symbolic statement into a numerical calculation. The left-hand side of the equation now involves multiplying -7 by -0.33. This multiplication is a key step in verifying our solution. We need to perform this calculation accurately to determine if it indeed equals the right-hand side of the equation, which is 2.31. The act of substitution is a powerful tool in mathematics. It allows us to test our solutions and confirm their validity. It also reinforces the understanding of how variables and equations work together. By substituting the solution back into the original equation, we are essentially reversing the steps we took to solve it. If the equation holds true after the substitution, it provides strong evidence that our solution is correct. Now, let's carry out the multiplication on the left-hand side and compare the result with the right-hand side.

Simplify the left-hand side

Simplifying the left-hand side, we have:

2. 31 = 2.31

This result confirms that our solution, y = -0.33, is correct. When we multiply -7 by -0.33, we obtain 2.31, which is exactly the value on the right-hand side of the original equation. This equality demonstrates that our solution satisfies the equation, meaning that it makes the equation a true statement. The verification process is now complete, and we can confidently assert that -0.33 is the solution to the equation -7y = 2.31. This final step underscores the importance of checking your work in mathematics. It provides a sense of closure and ensures that the answer you have obtained is accurate. The ability to verify solutions is a valuable skill that can be applied to a wide range of mathematical problems. It not only helps to prevent errors but also reinforces the understanding of the underlying concepts. With the verification complete, we have successfully solved the equation and confirmed our answer.

Conclusion

In conclusion, we have successfully solved the equation -7y = 2.31 by systematically isolating the variable 'y' and performing the necessary arithmetic operations. The solution we found is y = -0.33, which we verified by substituting it back into the original equation. This process highlights the fundamental principles of solving linear equations, including maintaining the balance of the equation and performing inverse operations to isolate the variable. The step-by-step approach we followed provides a clear and methodical way to tackle similar problems. By understanding and applying these principles, you can confidently solve a wide range of linear equations. Furthermore, the verification step emphasizes the importance of checking your work to ensure accuracy. This practice not only helps to prevent errors but also reinforces your understanding of the solution process. Solving equations is a core skill in mathematics, and mastering it opens the door to more advanced concepts and applications. Whether you are solving equations in an academic setting or applying them to real-world problems, the ability to manipulate and solve equations is an invaluable asset. The journey through this problem has not only provided us with a solution but also reinforced our understanding of algebraic principles and problem-solving strategies. With these skills in hand, you are well-equipped to tackle future mathematical challenges.