Solving For W In -231 = (7/11)w A Step-by-Step Guide

In the realm of mathematics, solving for an unknown variable is a fundamental skill. It allows us to unravel equations and find the specific value that makes the equation true. This article delves into a step-by-step solution for the equation -231 = (7/11)w, where our goal is to isolate the variable 'w' and determine its numerical value. This equation is a classic example of a linear equation in one variable, and mastering its solution technique lays the groundwork for tackling more complex algebraic problems. Understanding how to manipulate equations and isolate variables is crucial not only in mathematics but also in various fields like physics, engineering, and economics. Let's embark on this mathematical journey and discover the value of 'w' that satisfies the given equation.

Step 1: Isolating the Variable

The primary objective in solving for 'w' is to isolate it on one side of the equation. In the given equation, -231 = (7/11)w, 'w' is currently being multiplied by the fraction 7/11. To isolate 'w', we need to perform the inverse operation, which is division. However, dividing by a fraction can be a bit tricky. Instead, we can multiply both sides of the equation by the reciprocal of the fraction. The reciprocal of 7/11 is 11/7. Multiplying both sides by this reciprocal will effectively cancel out the fraction on the right side, leaving 'w' isolated.

This process utilizes the fundamental principle of equality, which states that performing the same operation on both sides of an equation maintains the equality. By multiplying both sides by 11/7, we ensure that the equation remains balanced, and we move closer to isolating 'w'. This step is a crucial application of algebraic manipulation and demonstrates the power of inverse operations in solving equations. In essence, we are undoing the multiplication by 7/11, thereby freeing 'w' from its coefficient.

Step 2: Multiplying by the Reciprocal

Now, let's execute the multiplication by the reciprocal. We multiply both sides of the equation -231 = (7/11)w by 11/7. This gives us:

(11/7) * (-231) = (11/7) * (7/11)w

On the left side, we have the product of a fraction and a negative integer. To simplify this, we can multiply the numerator (11) by -231 and then divide by the denominator (7). On the right side, the fraction 11/7 multiplied by its reciprocal 7/11 equals 1. This is because the numerator and denominator cancel each other out, leaving us with just 'w'.

This step highlights the elegance of using reciprocals to eliminate fractions in equations. It simplifies the equation and allows us to focus on the remaining arithmetic operations. The careful application of multiplication and cancellation is essential in this step, ensuring that we arrive at the correct value for 'w'. By strategically using the reciprocal, we've transformed the equation into a more manageable form, paving the way for the final solution.

Step 3: Simplifying the Equation

After multiplying both sides by the reciprocal, our equation now looks like this:

(11/7) * (-231) = w

To simplify the left side, we need to perform the multiplication. We can start by dividing -231 by 7. This gives us -33. Then, we multiply -33 by 11, which results in -363. Therefore, the left side of the equation simplifies to -363. This step involves basic arithmetic operations, but it's crucial to perform them accurately to arrive at the correct answer. Careful attention to signs (positive and negative) is particularly important in this step.

This simplification process demonstrates the importance of reducing complex expressions to their simplest forms. By performing the multiplication and division, we've condensed the left side of the equation into a single numerical value. This makes it much easier to compare and interpret the result. The ability to simplify equations is a fundamental skill in algebra and is essential for solving a wide range of mathematical problems.

Step 4: The Solution

After simplifying the equation, we arrive at the solution:

w = -363

This means that the value of 'w' that satisfies the original equation -231 = (7/11)w is -363. We have successfully isolated 'w' and determined its numerical value. This solution can be verified by substituting -363 back into the original equation and checking if both sides are equal. This step provides a final confirmation that our solution is correct and that we have accurately solved the equation.

This solution represents the culmination of our step-by-step process. We started with a complex equation and, through careful manipulation and simplification, arrived at a clear and concise answer. The value of w = -363 is the unique solution to the equation and demonstrates the power of algebraic techniques in solving for unknown variables. This process reinforces the importance of each step and the accuracy required to reach a correct solution.

Step 5: Verification (Optional but Recommended)

To ensure our solution is correct, we can substitute w = -363 back into the original equation:

-231 = (7/11) * (-363)

Now, we need to evaluate the right side of the equation. First, we multiply 7 by -363, which gives us -2541. Then, we divide -2541 by 11, which results in -231. Therefore, the right side of the equation simplifies to -231. Comparing this to the left side, we see that both sides are equal:

-231 = -231

This confirms that our solution w = -363 is indeed correct. Verification is a crucial step in problem-solving, especially in mathematics. It provides a safety net, ensuring that we haven't made any errors in our calculations or manipulations. By substituting the solution back into the original equation, we can gain confidence in our answer and ensure its accuracy. This step is a testament to the importance of thoroughness and attention to detail in mathematical problem-solving.

Conclusion

In conclusion, we have successfully solved the equation -231 = (7/11)w for 'w'. By following a step-by-step process of isolating the variable, multiplying by the reciprocal, simplifying the equation, and verifying the solution, we have determined that w = -363. This process highlights the fundamental principles of algebra and the importance of careful manipulation and simplification in solving equations. The ability to solve for unknown variables is a crucial skill in mathematics and is applicable in various real-world scenarios.

This journey through the equation demonstrates the power of algebraic techniques in unraveling mathematical problems. Each step, from isolating the variable to verifying the solution, plays a crucial role in the overall process. The value of w = -363 is not just a numerical answer; it represents the solution that satisfies the given equation and showcases the elegance and precision of mathematical reasoning. By mastering these techniques, we can confidently tackle more complex equations and apply these skills to a wide range of problems.

Therefore, the correct answer is C. w = -363