Solving For W In The Equation 3(w-8)-4=-3(-6w+4)-9w
Understanding the Equation
At the heart of this mathematical challenge lies the equation 3(w-8)-4=-3(-6w+4)-9w. To solve for w, we must embark on a journey of simplification and strategic manipulation. This equation, at first glance, may appear complex, but with a methodical approach, we can unravel its intricacies and reveal the value of w. The key lies in understanding the order of operations and applying algebraic principles to isolate the variable. Our initial focus will be on distributing the constants and eliminating parentheses, paving the way for combining like terms and ultimately isolating w on one side of the equation. This process involves a careful dance of addition, subtraction, multiplication, and division, all guided by the fundamental rules of algebra. As we navigate through the steps, we will meticulously maintain the balance of the equation, ensuring that any operation performed on one side is mirrored on the other. This delicate equilibrium is crucial for preserving the integrity of the equation and arriving at the correct solution. Furthermore, we will be vigilant in our calculations, double-checking each step to minimize the risk of errors. Solving for w is not merely about finding a numerical answer; it is about demonstrating a mastery of algebraic techniques and a commitment to precision. This equation serves as a microcosm of the broader mathematical landscape, where problem-solving skills are paramount and attention to detail is non-negotiable. By successfully tackling this challenge, we not only discover the value of w but also reinforce our understanding of fundamental mathematical principles. The journey may be challenging, but the reward of a solved equation and a strengthened grasp of algebra is well worth the effort.
Step-by-Step Solution
1. Distribute Constants
The initial step in solving for w involves distributing the constants on both sides of the equation. This process eliminates the parentheses and allows us to combine like terms more easily. On the left side, we distribute the 3 across the terms within the parentheses: 3 * w = 3w and 3 * -8 = -24. Similarly, on the right side, we distribute the -3 across the terms within the parentheses: -3 * -6w = 18w and -3 * 4 = -12. Applying these distributions, the equation transforms from 3(w-8)-4=-3(-6w+4)-9w to 3w - 24 - 4 = 18w - 12 - 9w. This seemingly small step is crucial as it unravels the complexity of the equation and sets the stage for further simplification. The distribution process is a fundamental algebraic operation that relies on the distributive property, which states that a(b + c) = ab + ac. By applying this property correctly, we ensure that each term within the parentheses is properly accounted for, maintaining the balance and integrity of the equation. It's essential to pay close attention to the signs when distributing, as a simple error in sign can lead to an incorrect solution. The careful distribution of constants is the first key to unlocking the value of w and navigating the intricate pathways of this equation.
2. Combine Like Terms
After distributing the constants, our next objective is to combine like terms on both sides of the equation. This simplification process streamlines the equation, making it easier to isolate w. On the left side, we identify the constant terms -24 and -4, which can be combined to yield -28. On the right side, we have two terms containing w: 18w and -9w. Combining these terms results in 9w. The equation now transforms from 3w - 24 - 4 = 18w - 12 - 9w to 3w - 28 = 9w - 12. Combining like terms is a fundamental algebraic technique that simplifies expressions by grouping together terms with the same variable or constant. This process relies on the commutative and associative properties of addition, which allow us to rearrange and regroup terms without altering the value of the expression. By combining like terms, we reduce the number of terms in the equation, making it more manageable and less prone to errors. This step is crucial in our quest to isolate w and solve the equation. It's essential to carefully identify and combine like terms, ensuring that we are adding or subtracting the coefficients correctly. The successful combination of like terms brings us closer to our goal of unraveling the value of w.
3. Isolate the Variable
To isolate the variable w, we need to strategically manipulate the equation to get all terms containing w on one side and all constant terms on the other. A common approach is to move the term with the smaller coefficient of w to the other side. In our equation, 3w - 28 = 9w - 12, the coefficient of w on the left side is 3, while on the right side it is 9. Therefore, we will subtract 3w from both sides to move the w term from the left to the right. This operation results in -28 = 6w - 12. Next, we need to isolate the constant term on the left side. To do this, we add 12 to both sides of the equation, which gives us -16 = 6w. Isolating the variable is a crucial step in solving any algebraic equation. It involves performing operations on both sides of the equation to move all terms containing the variable to one side and all constant terms to the other. This process relies on the addition and subtraction properties of equality, which state that adding or subtracting the same value from both sides of an equation does not change its solution. By strategically adding or subtracting terms, we can gradually peel away the layers of the equation and reveal the value of the variable. The key is to perform the same operation on both sides, maintaining the balance and integrity of the equation. As we isolate w, we are one step closer to discovering its numerical value.
4. Solve for w
With the equation now in the form -16 = 6w, the final step is to solve for w by dividing both sides by the coefficient of w, which is 6. Dividing both sides by 6, we get w = -16/6. This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This simplification yields w = -8/3. Therefore, the solution to the equation is w = -8/3. Solving for w is the culmination of our efforts, the moment when we finally unveil the numerical value of the variable. This final step involves performing the inverse operation of multiplication, which is division. By dividing both sides of the equation by the coefficient of w, we isolate w and determine its value. The solution, w = -8/3, represents the specific value that satisfies the original equation. This means that if we substitute -8/3 for w in the original equation, both sides of the equation will be equal. The process of solving for w demonstrates the power of algebraic manipulation and the importance of following a logical sequence of steps. From distributing constants to combining like terms to isolating the variable, each step plays a crucial role in unraveling the solution. With w now solved, we have successfully navigated the equation and conquered the mathematical challenge.
Final Answer
Therefore, the simplified solution for w is:
w = -8/3
This solution represents the value of w that satisfies the original equation, 3(w-8)-4=-3(-6w+4)-9w. We arrived at this answer through a series of meticulous steps, including distributing constants, combining like terms, isolating the variable, and finally, solving for w. Each step was carefully executed to maintain the balance and integrity of the equation, ensuring that our final answer is accurate and reliable. The solution w = -8/3 can be expressed as an improper fraction, highlighting its precise value. It's important to note that this solution can also be represented as a mixed number, -2 2/3, or as a decimal approximation, approximately -2.67. However, the fractional form, -8/3, is often preferred as it preserves the exact value of the solution. The successful determination of w = -8/3 demonstrates a strong understanding of algebraic principles and a mastery of problem-solving techniques. This solution not only answers the question at hand but also reinforces the importance of methodical approaches and attention to detail in mathematics. With w now solved, we have completed the journey of unraveling this equation and arrived at our final destination: the value of w.
