Solving For W A Step By Step Guide To -5(5w-19)=-19w-19
In this article, we will delve into solving the linear equation -5(5w - 19) = -19w - 19 for the variable 'w'. Linear equations are fundamental in mathematics, and mastering their solutions is crucial for various applications in science, engineering, and economics. This guide provides a step-by-step approach, ensuring clarity and understanding for learners of all levels. We will explore the underlying principles, perform algebraic manipulations, and arrive at the final solution. By the end of this article, you will have a solid grasp of how to solve this specific equation and similar linear equations.
Understanding Linear Equations
Before diving into the solution, let's briefly discuss what linear equations are and why they are important. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The variable is raised to the power of one, hence the term 'linear.' Linear equations can be written in the general form ax + b = 0, where x is the variable, and a and b are constants. Solving a linear equation involves finding the value of the variable that makes the equation true.
Linear equations are essential because they model many real-world scenarios. From calculating distances and speeds to determining costs and profits, linear equations provide a powerful tool for problem-solving. Proficiency in solving linear equations is also a stepping stone to more advanced mathematical concepts, such as systems of equations, inequalities, and calculus. Therefore, understanding the methods and techniques to solve these equations is crucial for both academic and practical purposes.
The Given Equation: -5(5w - 19) = -19w - 19
Our primary task is to solve the equation -5(5w - 19) = -19w - 19 for 'w'. This equation is a typical linear equation with the variable 'w' on both sides. To find the value of 'w', we need to isolate it on one side of the equation. This involves performing a series of algebraic operations, such as distribution, combining like terms, and applying inverse operations.
The process begins with careful attention to the order of operations and the properties of equality. Each step must be executed precisely to maintain the balance of the equation and arrive at the correct solution. In the following sections, we will break down each step in detail, providing explanations and justifications along the way. This methodical approach ensures that you not only understand the solution but also the reasoning behind each manipulation.
1. Distribute -5 into the Parentheses
The first step in solving the equation -5(5w - 19) = -19w - 19 is to distribute the -5 across the terms inside the parentheses. This means multiplying -5 by both 5w and -19. The distributive property states that a(b + c) = ab + ac. Applying this property, we get:
-5 * (5w) + -5 * (-19) = -25w + 95
So, the equation becomes:
-25w + 95 = -19w - 19
This step is crucial because it eliminates the parentheses and simplifies the equation, making it easier to work with. Accurate distribution is essential to maintaining the equality and ensuring the correct solution. Understanding the distributive property and its application is a fundamental skill in algebra, and this step provides a clear example of its use.
2. Move the Terms with 'w' to One Side
Now that we have -25w + 95 = -19w - 19, the next step is to move all the terms containing 'w' to one side of the equation. Conventionally, we move the terms to the side that results in a positive coefficient for 'w' to avoid dealing with negative signs unnecessarily. In this case, we can add 25w to both sides of the equation:
-25w + 95 + 25w = -19w - 19 + 25w
This simplifies to:
95 = 6w - 19
By adding 25w to both sides, we have eliminated the 'w' term from the left side and combined it with the 'w' term on the right side. This step is a key part of isolating the variable 'w'. The addition property of equality, which allows us to add the same value to both sides of an equation without changing its balance, is the principle behind this step.
3. Move the Constant Term to the Other Side
After moving the 'w' terms, we have 95 = 6w - 19. The next step is to move the constant term (-19) to the other side of the equation. To do this, we add 19 to both sides:
95 + 19 = 6w - 19 + 19
This simplifies to:
114 = 6w
By adding 19 to both sides, we have isolated the term with 'w' on the right side. This step is another application of the addition property of equality. The goal here is to separate the variable term from the constant terms, bringing us closer to isolating 'w' and finding its value.
4. Isolate 'w' by Dividing
Now we have 114 = 6w. To isolate 'w', we need to divide both sides of the equation by the coefficient of 'w', which is 6:
114 / 6 = (6w) / 6
This simplifies to:
w = 19
By dividing both sides by 6, we have successfully isolated 'w' and found its value. This step utilizes the division property of equality, which states that dividing both sides of an equation by the same non-zero number maintains the equality. This is the final algebraic manipulation needed to solve for 'w'.
5. Verify the Solution
To ensure our solution is correct, we should verify it by substituting w = 19 back into the original equation:
Original equation: -5(5w - 19) = -19w - 19
Substitute w = 19:
-5(5 * 19 - 19) = -19 * 19 - 19
Simplify:
-5(95 - 19) = -361 - 19
-5(76) = -380
-380 = -380
Since the left side of the equation equals the right side, our solution w = 19 is correct. Verification is a crucial step in solving equations as it confirms the accuracy of the solution and helps avoid errors. This process reinforces the understanding of equation solving and the properties of equality.
In this article, we have meticulously solved the linear equation -5(5w - 19) = -19w - 19 for the variable 'w'. Through a step-by-step approach, we distributed, combined like terms, and applied inverse operations to isolate 'w'. The solution we found is w = 19, which we verified by substituting it back into the original equation.
Solving linear equations is a fundamental skill in mathematics with wide-ranging applications. By mastering the techniques demonstrated in this guide, you can confidently tackle similar equations and build a strong foundation for more advanced mathematical concepts. The key takeaways include the importance of the distributive property, the properties of equality, and the methodical approach to isolating variables. Remember, practice is crucial for proficiency, so continue to solve various linear equations to strengthen your skills.
w = 19
