Solving 8(2w + 15) = 16w + 20 A Step-by-Step Guide

Solving linear equations is a fundamental skill in algebra. It involves finding the value of a variable that makes the equation true. In this article, we will walk through the process of solving the equation $8(2w + 15) = 16w + 20$ and determine the correct value for w. We will explore the steps involved in simplifying the equation, isolating the variable, and verifying the solution. This detailed explanation will not only provide a solution to this specific problem but also equip you with the knowledge to tackle similar equations with confidence. Understanding the underlying principles of equation solving is crucial for various mathematical and real-world applications. Whether you are a student learning algebra or someone looking to refresh your math skills, this guide offers a clear and concise approach to solving linear equations. We will also discuss common pitfalls and strategies to avoid them, ensuring a solid understanding of the process. The ability to solve linear equations is a building block for more advanced mathematical concepts, making it an essential skill to master. By the end of this article, you will have a comprehensive understanding of how to solve the equation $8(2w + 15) = 16w + 20$ and the broader principles of linear equation solving. The goal is to provide a step-by-step guide that is easy to follow and understand, regardless of your current mathematical background. So, let's dive in and start solving!

The Equation: $8(2w + 15) = 16w + 20$

Our primary task is to solve the equation $8(2w + 15) = 16w + 20$. This equation involves a variable, w, and our goal is to find the value of w that satisfies the equation. The equation is a linear equation, which means that the highest power of the variable is 1. Solving linear equations involves a series of algebraic manipulations to isolate the variable on one side of the equation. This process typically includes distributing, combining like terms, and using inverse operations to undo the operations performed on the variable. In this specific equation, we have a distributive property to apply on the left side and like terms to potentially combine after simplification. The equation also presents an opportunity to illustrate the importance of following the order of operations and performing each step correctly. Errors in algebraic manipulation can lead to incorrect solutions, so it's crucial to be meticulous and double-check each step. The equation $8(2w + 15) = 16w + 20$ is a classic example of a linear equation that can be solved using basic algebraic principles. By working through this equation step-by-step, we will demonstrate the key techniques involved in solving linear equations and highlight the importance of accuracy in the process. Understanding how to solve equations like this is essential for success in algebra and other areas of mathematics. It also has practical applications in various fields, such as physics, engineering, and economics, where mathematical models are used to represent real-world phenomena. So, let's proceed with the solution and uncover the value of w that makes the equation true.

Step 1: Distribute the 8

The first step in solving the equation $8(2w + 15) = 16w + 20$ is to distribute the 8 on the left side of the equation. This means we need to multiply 8 by each term inside the parentheses, which are $2w$ and $15$. The distributive property states that $a(b + c) = ab + ac$. Applying this property to our equation, we get: $8 * 2w + 8 * 15$. Multiplying 8 by $2w$ gives us $16w$, and multiplying 8 by 15 gives us 120. So, the left side of the equation becomes $16w + 120$. Now, our equation looks like this: $16w + 120 = 16w + 20$. Distributing correctly is a crucial step in solving equations, as it simplifies the equation and allows us to proceed with further algebraic manipulations. It's important to remember to multiply the term outside the parentheses by every term inside the parentheses. Failing to do so can lead to an incorrect equation and, consequently, an incorrect solution. In this case, distributing the 8 allows us to eliminate the parentheses and create a more manageable equation. This step sets the stage for combining like terms and isolating the variable. The distributive property is a fundamental concept in algebra, and mastering it is essential for solving a wide range of equations and mathematical problems. By carefully applying the distributive property, we have transformed our original equation into a simpler form, paving the way for the next steps in the solution process. Now, let's move on to the next step and see what we can do with our newly simplified equation.

Step 2: Simplify the Equation

After distributing the 8, our equation is now $16w + 120 = 16w + 20$. The next step in solving this equation is to simplify it by trying to isolate the variable w on one side. We can start by subtracting $16w$ from both sides of the equation. This will help us eliminate the w term from one side and see what remains. Subtracting $16w$ from both sides gives us: $16w + 120 - 16w = 16w + 20 - 16w$. On the left side, $16w - 16w$ cancels out, leaving us with 120. On the right side, $16w - 16w$ also cancels out, leaving us with 20. So, our equation now simplifies to: $120 = 20$. This is a significant result. We have eliminated the variable w entirely, and we are left with a statement that compares two constants. This simplified equation tells us something important about the original equation and its solutions. Simplifying equations is a crucial step in the solving process because it often reveals the nature of the solution. In this case, the simplification has led us to a contradiction, which means that the original equation has no solution. When solving equations, it's always a good idea to simplify as much as possible to make the equation easier to work with and to gain insights into the solution. By subtracting $16w$ from both sides, we have uncovered a fundamental property of the equation: it cannot be true for any value of w. This is a valuable piece of information that helps us conclude that there is no solution. Let's move on to the next step to formally state our conclusion.

Step 3: Analyze the Result

Our simplification led us to the equation $120 = 20$. This statement is clearly false. 120 is not equal to 20. This result indicates that the original equation, $8(2w + 15) = 16w + 20$, has no solution. In other words, there is no value of w that will make the equation true. When solving equations, if you arrive at a contradiction like this, it means that the equation is inconsistent. An inconsistent equation is one that has no solution. This is different from an identity, which is an equation that is true for all values of the variable. It's also different from a conditional equation, which is an equation that is true for some values of the variable but not others. The equation $120 = 20$ is a clear contradiction because it violates a fundamental mathematical truth. This contradiction arises because the coefficients and constants in the original equation are such that they lead to this impossible situation. Analyzing the result of our simplification is crucial because it allows us to draw meaningful conclusions about the original problem. In this case, the result tells us that we cannot find a value for w that satisfies the equation. This is an important finding, as it tells us that the problem has no solution within the realm of real numbers. Understanding how to analyze results like this is a valuable skill in algebra and mathematics in general. It allows us to not only solve equations but also to interpret the solutions (or lack thereof) in a meaningful way. Now, let's formally state our conclusion based on this analysis.

Conclusion: No Solution

Based on our step-by-step solution, we have determined that the equation $8(2w + 15) = 16w + 20$ has no solution. We arrived at this conclusion by distributing the 8, simplifying the equation, and ultimately obtaining the contradiction $120 = 20$. This contradiction indicates that there is no value of w that can make the original equation true. Therefore, the answer is "no solution". In summary, we started with the equation $8(2w + 15) = 16w + 20$, distributed the 8 to get $16w + 120 = 16w + 20$, subtracted $16w$ from both sides to simplify, and arrived at $120 = 20$. Since $120 = 20$ is a false statement, we concluded that the equation has no solution. This process demonstrates the importance of careful algebraic manipulation and the ability to interpret the results of those manipulations. When solving equations, it's crucial to check your work and be mindful of potential contradictions or inconsistencies. If you arrive at a contradiction, it means that the equation has no solution. This is a valid and important outcome in problem-solving. Understanding when an equation has no solution is just as important as knowing how to find a solution when one exists. By working through this example, we have reinforced the concepts of distribution, simplification, and analysis in equation solving. We have also highlighted the importance of recognizing and interpreting contradictions. The conclusion of "no solution" is a definitive answer to the problem, and it reflects our understanding of the underlying mathematical principles. We have successfully solved the problem by applying the appropriate algebraic techniques and interpreting the results correctly.

The final answer to the equation $8(2w + 15) = 16w + 20$ is no solution. This determination was made by following a series of algebraic steps, including distributing, simplifying, and analyzing the resulting equation. The key to arriving at this conclusion was recognizing the contradiction that arose when simplifying the equation. The equation $120 = 20$ is a false statement, which indicates that there is no value of w that can satisfy the original equation. This outcome is an important concept in algebra, as it demonstrates that not all equations have solutions. Some equations, like the one we solved, are inconsistent and have no solution. Understanding this concept is crucial for problem-solving in mathematics and other fields. The process of solving this equation reinforces the importance of careful algebraic manipulation and the ability to interpret the results of those manipulations. Each step, from distributing the 8 to simplifying the equation, was performed with precision to ensure an accurate outcome. The final answer of "no solution" is a testament to the rigorous application of algebraic principles and the ability to draw logical conclusions from mathematical results. This example serves as a valuable lesson in equation solving and the interpretation of mathematical statements. It highlights the fact that sometimes the answer is not a specific numerical value, but rather a statement about the nature of the equation itself. In this case, the statement is that the equation has no solution, which is a complete and correct answer to the problem.