Solving For W In -(4/7)w = 12 A Step-by-Step Guide

In the realm of mathematics, solving for an unknown variable is a fundamental skill. This article delves into the process of solving for w in the equation $-\frac{4}{7}w = 12$. This type of equation falls under the category of linear equations, which are a cornerstone of algebra. Mastering the techniques to solve these equations is crucial for success in higher-level mathematics and various real-world applications. We'll break down the steps involved, ensuring a clear understanding for learners of all levels. Understanding how to isolate a variable, like w in this case, is essential for various mathematical problems, including those encountered in physics, engineering, economics, and computer science. The ability to manipulate equations and solve for unknowns allows us to model real-world situations, make predictions, and design solutions to complex problems. This article will not only demonstrate the step-by-step solution to the given equation but also highlight the underlying principles and concepts that make this process universally applicable. By grasping these concepts, you'll be well-equipped to tackle a wide range of algebraic challenges. The importance of accuracy and precision in mathematical calculations cannot be overstated. A single error in a step can lead to an incorrect final answer. Therefore, it's crucial to pay close attention to detail and double-check each step of the solution process. Developing a methodical approach to problem-solving, where each step is clearly defined and executed, will greatly minimize the risk of errors. Furthermore, understanding the properties of equality, such as the multiplication property of equality, is vital for manipulating equations correctly. These properties allow us to perform operations on both sides of an equation without changing its fundamental balance. This ensures that the solutions we obtain are valid and accurate. So, let's embark on this journey of solving for w and unlock the power of algebraic equations!

Understanding the Equation -(4/7)w = 12

Before diving into the solution, let's dissect the equation $-\frac{4}{7}w = 12$. This is a linear equation in one variable, w. The coefficient of w is $-\frac{4}{7}$, and the constant term is 12. Our goal is to isolate w on one side of the equation to determine its value. The equation represents a relationship where a fraction multiplied by the variable w equals a constant. This type of relationship is commonly encountered in various mathematical and scientific contexts. For instance, it could represent the relationship between the amount of work done and the time taken, or the relationship between distance traveled and speed. Understanding the structure of the equation is crucial for choosing the appropriate solution strategy. In this case, we need to undo the multiplication by the fraction $-\frac{4}{7}$. This can be achieved by multiplying both sides of the equation by the reciprocal of the fraction. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. In the next section, we'll explore the detailed steps involved in isolating w and finding its value. It's important to remember that each step we take must maintain the equality of the equation. This means that whatever operation we perform on one side of the equation, we must also perform on the other side. This principle ensures that the equation remains balanced and that the solution we obtain is correct. The ability to interpret equations and identify the relationships they represent is a key skill in mathematics. By understanding the underlying structure of an equation, we can choose the most efficient method for solving it. This not only saves time but also reduces the risk of errors. So, let's proceed to the next step and see how we can isolate w and find its solution.

Step-by-Step Solution to Solve for w

The key to solving for w in the equation $-\frac{4}{7}w = 12$ lies in isolating w on one side of the equation. This can be achieved by multiplying both sides of the equation by the reciprocal of the coefficient of w. The coefficient of w is $-\frac{4}{7}$, and its reciprocal is $-\frac{7}{4}$. By multiplying both sides of the equation by $-\frac{7}{4}$, we can effectively cancel out the coefficient of w and isolate the variable. This method is based on the fundamental principle of algebraic manipulation, which states that performing the same operation on both sides of an equation maintains its equality. This principle is crucial for solving equations correctly and ensuring that the solution obtained is valid. Let's perform the multiplication step-by-step:

1. Multiply both sides by the reciprocal: (-\frac{7}{4}) * (-\frac{4}{7}w) = 12 * (-\frac{7}{4})

2. Simplify the left side: The negative signs cancel out, and the fractions multiply to 1, leaving us with: w = 12 * (-\frac{7}{4})

3. Simplify the right side: We can simplify the right side by multiplying 12 by the fraction $-\frac{7}{4}$. This can be done by first multiplying 12 by -7, which gives us -84, and then dividing by 4: w = -\frac{84}{4}

4. Further simplification: Finally, we divide -84 by 4 to obtain the value of w: w = -21

Therefore, the solution to the equation $-\frac{4}{7}w = 12$ is w = -21. This solution can be verified by substituting -21 back into the original equation and checking if the equation holds true. This step is crucial for ensuring the correctness of the solution and identifying any potential errors in the calculation. In the next section, we'll discuss the importance of verifying the solution and demonstrate the verification process for this equation. So, let's proceed to the next step and solidify our understanding of the solution process.

Verifying the Solution: Ensuring Accuracy

After obtaining a solution to an equation, it's crucial to verify its accuracy. This step ensures that the solution is correct and that no errors were made during the solving process. Verification involves substituting the obtained value back into the original equation and checking if both sides of the equation are equal. This process helps to identify any mistakes in the calculations and provides confidence in the solution. For the equation $-\frac{4}{7}w = 12$, we found that w = -21. To verify this solution, we substitute -21 back into the original equation:

-\frac{4}{7} * (-21) = 12

Let's simplify the left side of the equation:

-\frac{4}{7} * (-21) = \frac{4 * 21}{7} = \frac{84}{7} = 12

As we can see, the left side of the equation simplifies to 12, which is equal to the right side of the equation. This confirms that our solution, w = -21, is correct. The verification process not only ensures the accuracy of the solution but also reinforces the understanding of the equation and the solution process. It's a valuable step in problem-solving that helps to build confidence and develop a deeper understanding of mathematical concepts. In more complex equations, verification can be more challenging, but the principle remains the same: substitute the solution back into the original equation and check if it holds true. This step is particularly important when dealing with equations that involve fractions, decimals, or other complex operations. By verifying the solution, we can be certain that we have arrived at the correct answer and that our understanding of the problem is accurate. So, always remember to verify your solutions to ensure accuracy and build confidence in your mathematical abilities.

Conclusion: Mastering Algebraic Equations

In conclusion, solving for w in the equation $-\frac{4}{7}w = 12$ demonstrates a fundamental algebraic skill – isolating a variable. By multiplying both sides of the equation by the reciprocal of the coefficient of w, we successfully found that w = -21. Furthermore, we emphasized the importance of verifying the solution by substituting it back into the original equation. This process ensures the accuracy of the solution and reinforces the understanding of the equation. Mastering these techniques is crucial for success in mathematics and various other fields. The ability to solve algebraic equations is a valuable skill that can be applied to a wide range of problems, from simple arithmetic to complex scientific calculations. By understanding the principles of algebraic manipulation and practicing regularly, anyone can develop proficiency in solving equations. Remember that each step in the solution process is important, and accuracy is paramount. Pay close attention to detail, and always double-check your work. Verification is a powerful tool that can help you identify and correct errors, ensuring that your solutions are correct. As you continue your mathematical journey, you will encounter more complex equations and problems. However, the fundamental principles and techniques learned here will serve as a solid foundation for your future success. So, keep practicing, keep learning, and keep exploring the fascinating world of mathematics! The journey of learning mathematics is a continuous one, and each problem solved is a step forward. With dedication and perseverance, you can master algebraic equations and unlock the power of mathematical problem-solving.