Solving W/8 + 3.1 = -22.5: A Step-by-Step Guide

Are you struggling with the equation w/8 + 3.1 = -22.5? Don't worry, you're not alone! Many people find algebra a bit tricky, but with a clear understanding of the steps involved, you can easily solve this type of equation. This comprehensive guide will walk you through each step, providing clear explanations and helpful tips along the way. By the end of this article, you'll not only know how to solve this specific equation, but you'll also have a better grasp of the fundamental principles of algebra, empowering you to tackle similar problems with confidence. Let's dive in and unlock the solution together!

Understanding the Equation

Before we jump into solving, let's take a moment to understand what the equation w/8 + 3.1 = -22.5 actually means. In algebra, an equation is a statement that two expressions are equal. In this case, we have an expression on the left side of the equals sign (w/8 + 3.1) and an expression on the right side (-22.5). Our goal is to find the value of the variable w that makes this equation true. The variable w represents an unknown number, and our task is to isolate w on one side of the equation to determine its value. To do this, we'll use the principles of inverse operations, which involve performing the opposite operation to undo a mathematical operation.

Think of it like a puzzle where we need to carefully manipulate the equation while maintaining its balance. Each step we take must preserve the equality, ensuring that both sides of the equation remain equal. This requires a methodical approach and a clear understanding of the order of operations. Remember, the key to success in algebra is to break down complex problems into smaller, more manageable steps. So, let's start by addressing the constant term on the left side of the equation and working our way towards isolating w. With patience and practice, you'll find that solving equations like this becomes second nature. Now, let's move on to the first step in our solution: isolating the term with w.

Step 1: Isolate the Term with 'w'

Our first goal in solving the equation w/8 + 3.1 = -22.5 is to isolate the term that contains the variable w, which in this case is w/8. To do this, we need to get rid of the + 3.1 on the left side of the equation. We can achieve this by using the inverse operation of addition, which is subtraction. The principle here is that whatever operation we perform on one side of the equation, we must also perform on the other side to maintain the equality. This ensures that the equation remains balanced and that we are not changing the solution.

So, we will subtract 3.1 from both sides of the equation. This can be written as:

w/8 + 3.1 - 3.1 = -22.5 - 3.1

On the left side, the + 3.1 and - 3.1 cancel each other out, leaving us with just w/8. On the right side, we perform the subtraction -22.5 - 3.1, which equals -25.6. Therefore, our equation now looks like this:

w/8 = -25.6

We have successfully isolated the term with w on the left side of the equation. This is a significant step forward because we are now closer to finding the value of w. The next step will involve dealing with the division by 8. Remember, the key to isolating w is to systematically undo the operations that are being applied to it. By subtracting 3.1 from both sides, we eliminated the addition. Now, we need to address the division. Are you ready to move on to the next step and see how we can get w all by itself? Let's do it!

Step 2: Solve for 'w'

Now that we have the equation w/8 = -25.6, our next task is to solve for w. Currently, w is being divided by 8. To isolate w, we need to perform the inverse operation of division, which is multiplication. We'll multiply both sides of the equation by 8. Remember, whatever we do to one side of the equation, we must do to the other side to maintain balance and ensure the equality remains true.

So, we multiply both sides of the equation by 8:

(w/8) * 8 = -25.6 * 8

On the left side, multiplying w/8 by 8 cancels out the division, leaving us with just w. On the right side, we perform the multiplication -25.6 * 8. This calculation results in -204.8.

Therefore, our equation now simplifies to:

w = -204.8

We have successfully solved for w! This means that the value of w that makes the original equation true is -204.8. We have isolated w and found its value by using the principles of inverse operations. To be completely sure of our answer, it's always a good idea to check our solution by plugging it back into the original equation. Let's move on to the final step, which involves verifying our solution.

Step 3: Verify the Solution

To ensure that our solution w = -204.8 is correct, we need to substitute this value back into the original equation w/8 + 3.1 = -22.5. This process is called verifying the solution, and it's a crucial step in problem-solving. By plugging in our calculated value, we can check if both sides of the equation are indeed equal. If they are, then our solution is correct. If not, then we need to go back and review our steps to identify any potential errors.

Let's substitute w = -204.8 into the original equation:

(-204.8) / 8 + 3.1 = -22.5

First, we perform the division: (-204.8) / 8 = -25.6

Now, we have:

-25.6 + 3.1 = -22.5

Next, we perform the addition: -25.6 + 3.1 = -22.5

So, our equation now reads:

-22.5 = -22.5

Since both sides of the equation are equal, this confirms that our solution w = -204.8 is correct. We have successfully verified our answer by substituting it back into the original equation and showing that it satisfies the equality. This process provides us with confidence in our solution and ensures that we have accurately solved the problem. Now that we've walked through the entire process, let's summarize the steps we took and highlight the key concepts involved.

Conclusion

In this comprehensive guide, we've successfully solved the equation w/8 + 3.1 = -22.5 step by step. We started by understanding the equation and identifying our goal: to isolate the variable w. We then used the principles of inverse operations to systematically undo the operations being applied to w. First, we subtracted 3.1 from both sides of the equation to isolate the term with w, resulting in w/8 = -25.6. Next, we multiplied both sides by 8 to solve for w, which gave us the solution w = -204.8. Finally, we verified our solution by substituting it back into the original equation, confirming that both sides were equal and that our answer was correct.

By following these steps, you can confidently solve similar algebraic equations. Remember, the key is to understand the principles of inverse operations and to maintain balance by performing the same operations on both sides of the equation. With practice, you'll become more comfortable and proficient in solving algebraic problems. This process not only helps in solving mathematical problems but also enhances logical thinking and problem-solving skills that are valuable in various aspects of life. Keep practicing and exploring different types of equations to further develop your algebraic skills. For more resources and practice problems, you can visit websites like Khan Academy Algebra, which offers a wealth of tutorials and exercises to help you master algebra.