Solving 8(4s + 2) = 10s + 16 A Step By Step Guide

Introduction

In this article, we will delve into the process of solving the algebraic equation $\begin{aligned} 8(4s+2) & =10s+16 \\ 32s+\square & =10s+16 \\ \square+16 & =16 \\ 22s & =\square \\ s & =\square\end{aligned}$. Solving equations is a fundamental skill in mathematics, and understanding the step-by-step methodology is crucial for success in algebra and beyond. This guide aims to provide a comprehensive explanation of each step involved in solving the given equation, ensuring that you grasp the underlying principles and can apply them to similar problems. We will break down the equation into manageable parts, explaining the logic behind each operation. From the initial distribution to isolating the variable 's', we will cover every aspect in detail. By the end of this article, you should have a clear understanding of how to solve this equation and similar algebraic problems. The importance of algebraic equations in various fields, including engineering, physics, and economics, cannot be overstated. Mastering these skills will not only help you in your academic pursuits but also in real-world problem-solving. Our approach will focus on clarity and simplicity, making the concepts accessible to learners of all levels. We will emphasize the importance of accuracy in each step and demonstrate how a systematic approach can prevent errors. So, let’s begin our journey into the world of algebra and learn how to solve this equation effectively.

Step 1: Distribute the 8

The first step in solving the equation $\begin{aligned} 8(4s+2) & =10s+16 \end{aligned}$ is to distribute the 8 across the terms inside the parentheses. This involves multiplying 8 by both 4s and 2. Distribution is a key algebraic technique that simplifies equations by removing parentheses. When we multiply 8 by 4s, we get 32s. Similarly, when we multiply 8 by 2, we get 16. Therefore, the left side of the equation becomes 32s + 16. It’s crucial to perform this step accurately to avoid errors in the subsequent steps. Miscalculations at this stage can lead to an incorrect final answer. So, let’s reiterate the process: 8 multiplied by 4s equals 32s, and 8 multiplied by 2 equals 16. Consequently, the equation now looks like this: 32s + 16 = 10s + 16. Understanding the distributive property is essential for solving more complex equations as well. This property states that a(b + c) = ab + ac. In our case, a is 8, b is 4s, and c is 2. By applying this property, we've successfully expanded the left side of the equation. This expansion allows us to combine like terms and eventually isolate the variable 's'. Remember, the goal is to simplify the equation step by step until we can determine the value of 's'. Accuracy in this distribution step is paramount, so double-check your calculations to ensure you're on the right track. Now that we've completed the distribution, we can move on to the next phase of solving the equation.

Thus, we fill in the first blank:

$\begin{aligned} 8(4s+2) & =10s+16 \\ 32s+16 & =10s+16 \\ \square+16 & =16 \\ 22s & =\square \\ s & =\square\end{aligned}$

Step 2: Isolate the Variable Terms

After distributing the 8, our equation stands as 32s + 16 = 10s + 16. The next step involves isolating the variable terms. This means we want to gather all terms containing 's' on one side of the equation. To achieve this, we subtract 10s from both sides of the equation. Subtracting the same value from both sides maintains the equality, which is a fundamental principle in algebra. When we subtract 10s from 32s, we get 22s. On the right side, 10s - 10s cancels out, leaving us with just the constant term. Therefore, the equation now becomes 22s + 16 = 16. The reason for isolating the variable terms is to simplify the equation further, bringing us closer to finding the value of 's'. By moving all 's' terms to one side, we can eventually isolate 's' itself. This process is a cornerstone of solving algebraic equations. The key here is to perform the same operation on both sides of the equation to maintain balance. If we were to subtract 10s from only one side, the equation would no longer be valid. Understanding the concept of maintaining equality is vital for successful equation solving. This principle applies to all algebraic manipulations, including addition, subtraction, multiplication, and division. So, by subtracting 10s from both sides, we've made significant progress in isolating the variable. We’ve successfully moved the 's' terms to the left side of the equation, setting the stage for the next step, which will involve isolating 's' even further. This methodical approach is what makes algebra solvable, even for complex equations.

Step 3: Isolate the Constant Terms

Following the isolation of variable terms, our equation is 22s + 16 = 16. Now, we need to isolate the constant terms. This involves moving all the numerical values (constants) to one side of the equation, which in this case will be the right side. To do this, we subtract 16 from both sides of the equation. Remember, maintaining equality is crucial, so whatever operation we perform on one side, we must also perform on the other. Subtracting 16 from the left side cancels out the +16, leaving us with just 22s. On the right side, 16 - 16 equals 0. Thus, our equation simplifies to 22s = 0. This step is essential because it further simplifies the equation, making it easier to solve for 's'. By isolating the constants, we’ve effectively separated the variable term from the numerical values. This separation allows us to perform the final step of dividing by the coefficient of 's'. The logic behind isolating constants is similar to that of isolating variables. We aim to simplify the equation by grouping like terms together. This process makes the equation more manageable and reduces the chances of making errors. The subtraction of 16 from both sides is a direct application of the additive inverse property, which states that adding a number and its inverse results in zero. In this case, 16 and -16 are additive inverses. By understanding and applying these fundamental algebraic principles, we can confidently solve a wide range of equations. The equation 22s = 0 represents a significant milestone in our solving process. We are now just one step away from determining the value of 's'. The next and final step will involve dividing both sides of the equation by 22.

Thus, we fill in the second blank:

$\begin{aligned} 8(4s+2) & =10s+16 \\ 32s+16 & =10s+16 \\ 22s+16 & =16 \\ 22s & =\square \\ s & =\square\end{aligned}$

Step 4: Solve for 's'

Our equation now reads 22s = 0. The final step is to solve for 's'. To do this, we need to isolate 's' completely. Since 's' is being multiplied by 22, we perform the inverse operation, which is division. We divide both sides of the equation by 22. Dividing both sides by the same number maintains the equality, a principle we’ve emphasized throughout this guide. When we divide 22s by 22, we are left with just 's'. On the right side, we have 0 divided by 22. Any number divided by zero is zero. Therefore, s = 0. This is our solution to the equation. We have successfully isolated 's' and determined its value. The act of solving for the variable is the culmination of all the previous steps. Each step, from distribution to isolating terms, was a means to this end. The solution s = 0 means that if we substitute 0 for 's' in the original equation, both sides of the equation will be equal. This is a way to check our answer and ensure we haven't made any errors. The division step is a crucial one in algebra. It allows us to undo the multiplication that is affecting the variable. Without this step, we would not be able to determine the value of 's'. The principle of using inverse operations is fundamental to solving equations. We use addition to undo subtraction, subtraction to undo addition, multiplication to undo division, and division to undo multiplication. By mastering this principle, you can tackle a wide variety of algebraic problems. We have now completed all the steps necessary to solve the equation. The solution s = 0 is the final answer. Understanding each step and the reasoning behind it is crucial for building a strong foundation in algebra.

Thus, we fill in the third and fourth blanks:

$\begin{aligned} 8(4s+2) & =10s+16 \\ 32s+16 & =10s+16 \\ 22s+16 & =16 \\ 22s & =0 \\ s & =0\end{aligned}$

Conclusion

In conclusion, we have successfully solved the equation $\begin{aligned} 8(4s+2) & =10s+16 \end{aligned}$ step by step. We began by distributing the 8 across the terms inside the parentheses, resulting in 32s + 16 = 10s + 16. Next, we isolated the variable terms by subtracting 10s from both sides, which gave us 22s + 16 = 16. Then, we isolated the constant terms by subtracting 16 from both sides, simplifying the equation to 22s = 0. Finally, we solved for 's' by dividing both sides by 22, which yielded the solution s = 0. This step-by-step process is a testament to the power of algebraic manipulation in solving equations. By following a systematic approach, we can break down complex problems into manageable steps and arrive at the correct solution. The key to success in algebra is understanding the underlying principles and applying them consistently. The principles of maintaining equality, using inverse operations, and simplifying expressions are fundamental to solving equations. The ability to solve algebraic equations is a valuable skill that extends beyond the classroom. It is essential for various fields, including science, engineering, finance, and computer science. Mastering these skills will not only enhance your academic performance but also equip you with the tools to solve real-world problems. We hope this guide has provided you with a clear and comprehensive understanding of how to solve this equation. Remember to practice regularly and apply these techniques to other problems to solidify your understanding. With consistent effort and a methodical approach, you can excel in algebra and unlock its many applications.