Solving Linear Equations: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a classic problem: solving the equation $\frac{w}{10} - 3.1 = 4.2$. It's a linear equation, and we'll break down the steps to find the value of w. Don't worry, it's not as scary as it looks. We'll go through it step by step, making sure you understand every move. Our main goal is to isolate the variable (w) and find its value. So, let's get started, shall we?

This kind of problem falls under the umbrella of algebra, specifically dealing with linear equations. These equations are fundamental in mathematics and have applications everywhere, from physics and engineering to economics and computer science. The key concept here is to manipulate the equation using mathematical operations (addition, subtraction, multiplication, and division) while keeping the equation balanced. Think of it like a seesaw; whatever you do on one side, you must do on the other to keep it balanced. Our approach here revolves around undoing the operations that have been applied to w until w is all alone on one side of the equation. We will be doing the reverse order of operations (PEMDAS/BODMAS). This is a foundational skill in algebra, and getting comfortable with it opens the door to tackling more complex mathematical problems down the road. Let's get our hands dirty, guys.

Step 1: Isolate the Term with w

Alright, first things first. Our equation is $\frac{w}{10} - 3.1 = 4.2$. We want to get the term with w ($\frac{w}{10}$) by itself on one side of the equation. To do that, we need to get rid of the -3.1. The opposite of subtracting 3.1 is adding 3.1. So, we'll add 3.1 to both sides of the equation. This maintains the equality. Remember, we need to keep the equation balanced.

So we do this: $\frac{w}{10} - 3.1 + 3.1 = 4.2 + 3.1$. On the left side, -3.1 and +3.1 cancel each other out, leaving us with $\frac{w}{10}$. On the right side, 4.2 + 3.1 equals 7.3. Therefore, our equation now simplifies to $\frac{w}{10} = 7.3$. We are making progress! We've successfully isolated the term containing our variable w. We're getting closer to solving the equation. Always remember to perform the same operation on both sides to keep the equation balanced. This is a very important concept in algebra. This step is about creating a simpler form of the equation.

Now, let's reflect on the importance of this step. Isolating the term with the variable is a fundamental technique in algebra. It's the first major step in solving any linear equation. By strategically adding or subtracting terms from both sides, we effectively simplify the equation, making it easier to solve. The aim is always to reduce the equation to a form where the variable is isolated and its value can be easily determined. Mastering this step lays a strong foundation for tackling more complex algebraic problems. Imagine you're trying to find a hidden treasure; this step is like removing the layers of sand to reveal the chest. The key is to recognize the operations performed on the variable and use the inverse operations to undo them. With practice, you'll become a pro at isolating terms. Keep up the good work!

Step 2: Solve for w

Okay, we've got $\frac{w}{10} = 7.3$. Now, we need to get w completely by itself. Currently, w is being divided by 10. The inverse operation of division is multiplication. So, we'll multiply both sides of the equation by 10. Again, this keeps the equation balanced.

So we perform: $10 * \frac{w}{10} = 7.3 * 10$. On the left side, the 10 in the numerator and the 10 in the denominator cancel each other out, leaving us with w. On the right side, 7.3 multiplied by 10 equals 73. Therefore, our equation now becomes w = 73. And there you have it, folks! We've solved for w! It seems that w equals 73. This is the solution to our equation. This is the final step in our journey.

This step brings us to the exciting moment of finding the solution. Multiplying both sides of the equation by 10 is the key to isolating w. This is a critical step because it removes the division by 10, leaving w as the subject of the equation. This concept of applying inverse operations is crucial in algebra. In essence, it's about reversing the operations applied to the variable to reveal its value. This step is like using the right key to unlock the treasure chest, revealing the solution we've been working towards. It's important to remember that whatever operation is performed on one side of the equation, it must also be performed on the other to maintain the equation's balance. This ensures that the solution obtained is valid. The aim is to get the variable by itself. This is the cornerstone of algebraic problem-solving, opening the door to tackling complex equations.

Step 3: Verify Your Answer

It's always a great idea to check your answer! To do this, substitute the value of w (which is 73) back into the original equation and see if it holds true. So, our original equation was $\frac{w}{10} - 3.1 = 4.2$. Let's plug in 73 for w:

$\frac{73}{10} - 3.1 = 4.2$. Now, $\frac{73}{10}$ is equal to 7.3. So the equation becomes: 7.3 - 3.1 = 4.2. And, indeed, 7.3 - 3.1 does equal 4.2. That means our answer, w = 73, is correct! Nice job, everyone!

Verification is an essential step in problem-solving. It confirms the accuracy of the solution. By substituting the found value of w back into the original equation, we are essentially testing whether it satisfies the equation. This check not only validates our work but also builds confidence in our problem-solving abilities. It's like double-checking your work on a test to ensure you didn't make any errors. This step is about ensuring the solution is accurate. This practice helps to reinforce the understanding of the equation and its components. If the equation holds true after substitution, we can be confident that our solution is correct. In cases where the solution does not satisfy the equation, it signals an error in our calculations, prompting us to revisit our steps and identify where we went wrong. This is the most crucial part because we double-check the results and make sure the answer is correct.

Conclusion: You Did It!

Congratulations, guys! You've successfully solved the linear equation $\frac{w}{10} - 3.1 = 4.2$. We went through the steps together, from isolating the term with the variable to solving for w and verifying our answer. Remember, the key is to perform inverse operations to isolate the variable and keep the equation balanced. Keep practicing, and you'll become a pro at solving linear equations. Math can be fun when you understand the steps. Keep going and keep learning! You are all awesome!

We successfully navigated the process of solving a linear equation, showcasing the power of basic algebraic principles. We've shown how the equation is manipulated to find the value of the unknown variable, w. Remember that understanding the fundamental concepts, such as inverse operations and maintaining equation balance, is crucial. This will help you in many mathematical problems. This methodical approach will benefit you in all mathematical endeavors. Keep practicing; the more you work through problems, the more confident you'll become. Each problem you solve is a step forward, building your problem-solving skills. So keep learning and enjoy the journey!