Solving Equations $9 - \frac{1}{4}x = \frac{1}{8}x$ A Step-by-Step Guide

In the realm of mathematics, solving for a variable is akin to detective work. We're presented with an equation, a mathematical puzzle, and our mission is to isolate the unknown, the variable, to uncover its value. In this guide, we'll embark on a step-by-step journey to solve the equation $9 - \frac{1}{4}x = \frac{1}{8}x$, and more importantly, we'll learn how to verify our solution, ensuring we've cracked the case correctly. This is a foundational skill in algebra, and mastering it opens doors to more complex mathematical concepts. When we talk about solving equations, we are essentially trying to find the value of the unknown variable that makes the equation true. This process involves manipulating the equation using algebraic principles until the variable is isolated on one side.

Think of an equation as a balanced scale. The equals sign (=) represents the fulcrum, and the expressions on either side represent the weights. To keep the scale balanced, any operation performed on one side must also be performed on the other. This principle is crucial for maintaining the equation's integrity as we solve for the variable. So, if you add something to the left side, you've got to add the exact same thing to the right side. It's like giving the scale a little nudge but making sure both sides still weigh the same. This is why balancing the equation is a key concept in algebra. When we say "solve for the variable," it means we're trying to get the variable, usually represented by x, all by itself on one side of the equation. The number or expression on the other side will then be the solution, telling us what value of x makes the equation true. It's like finding the secret code that unlocks the equation's meaning. The equation we're tackling today, $9 - \frac{1}{4}x = \frac{1}{8}x$, looks a little more complex with those fractions hanging around the x terms. But don't worry, guys! We'll break it down step-by-step and make it super easy to understand. We're going to use our algebraic skills to move those terms around, combine like terms, and eventually isolate x all by itself. And the best part? We're not just going to find the answer. We're going to check it too! That way, we know for sure that our solution is correct. Think of it like double-checking your work on a test – always a good idea!

Our equation is $9 - \frac{1}{4}x = \frac{1}{8}x$. The first hurdle we encounter is the fractions. Fractions in equations can sometimes feel like a bit of a headache, but there's a neat trick to get rid of them. We can multiply both sides of the equation by the least common multiple (LCM) of the denominators. The denominators here are 4 and 8. The LCM of 4 and 8 is 8. So, we'll multiply both sides of the equation by 8. This is a strategic move to eliminate the fractions and simplify the equation. Remember, whatever we do to one side, we must do to the other to maintain the balance. So, let's multiply both sides of the equation by 8: $8 * (9 - \frac{1}{4}x) = 8 * (\frac{1}{8}x)$. Now, we need to distribute the 8 on both sides. On the left side, we have $8 * 9$ and $8 * -\frac{1}{4}x$. On the right side, we have $8 * \frac{1}{8}x$. Let's perform these multiplications: $72 - 2x = x$. See how the fractions have vanished? We've successfully cleared the fractions, making the equation much easier to work with. This step is all about simplifying the equation. By multiplying both sides by the LCM, we've transformed a potentially messy equation into a cleaner, more manageable one. Now, we have an equation without fractions, which is a big win! The next step involves isolating the x terms. Currently, we have $-2x$ on the left side and x on the right side. To get all the x terms together, we can add $2x$ to both sides of the equation. This will eliminate the $-2x$ on the left side and move it to the right side. So, let's do that: $72 - 2x + 2x = x + 2x$. Simplifying this gives us: $72 = 3x$. We're getting closer! Now, we have all the x terms on one side and the constant term on the other. To finally isolate x, we need to divide both sides of the equation by the coefficient of x, which is 3. This will give us the value of x. So, let's divide both sides by 3: $\frac{72}{3} = \frac{3x}{3}$. This simplifies to: $24 = x$. Hooray! We've found our solution: $x = 24$. But we're not done yet. We need to check our answer to make sure it's correct. This is a crucial step in the problem-solving process.

Finding a solution is great, but knowing that your solution is correct is even better. That's why checking the solution is an indispensable step in solving equations. It's like having a built-in error detector. To check our solution, we substitute the value we found for x back into the original equation. If the equation holds true, then our solution is correct. If not, we know we need to go back and look for a mistake. So, let's take our solution, $x = 24$, and plug it back into the original equation: $9 - \frac{1}{4}x = \frac{1}{8}x$. Substituting $x = 24$, we get: $9 - \frac{1}{4}(24) = \frac{1}{8}(24)$. Now, we need to simplify both sides of the equation. On the left side, we have: $9 - \frac{1}{4}(24) = 9 - 6 = 3$. On the right side, we have: $\frac{1}{8}(24) = 3$. So, we have: $3 = 3$. This is a true statement! The left side equals the right side, which means our solution, $x = 24$, is correct. We've successfully solved the equation and verified our answer. This process of verifying the solution is so important because it gives us confidence in our answer. It's like having a final seal of approval on our work. Without checking, we might unknowingly carry an incorrect solution forward, which could lead to problems later on. Checking also helps us catch any arithmetic errors or mistakes in our algebraic manipulations. It's a safeguard against making simple mistakes that can derail our entire solution. So, always make sure to plug your solution back into the original equation and see if it works. If it does, you've nailed it! If not, it's a sign to revisit your steps and find the error. In our case, substituting $x = 24$ into the original equation gave us $3 = 3$, which confirms that our solution is indeed correct. We've not only solved for x, but we've also proven that our solution is accurate. That's a double win!

After our step-by-step journey through the equation $9 - \frac{1}{4}x = \frac{1}{8}x$, we've arrived at our destination: the solution. We meticulously cleared fractions, combined like terms, and isolated the variable x. Our efforts have paid off, and we've discovered that $x = 24$. This means that the value of x that makes the equation true is 24. But our journey didn't end with just finding the solution. We went the extra mile and checked our answer, ensuring its accuracy. By substituting $x = 24$ back into the original equation, we confirmed that both sides of the equation were equal, giving us the confidence that our solution is correct. So, the final answer is: $x = 24$. This is the value that, when plugged into the equation, makes the statement true. Understanding the solution in the context of the original equation is crucial. It's not just about getting a number; it's about understanding what that number represents and how it satisfies the equation. In this case, 24 is the value of x that balances the equation, making both sides equal. This concept of balancing equations is fundamental in algebra, and it's the key to solving for unknowns in a variety of mathematical problems. The significance of the solution extends beyond just this particular equation. The skills we've honed in solving this equation – clearing fractions, combining like terms, isolating variables, and checking solutions – are transferable to many other algebraic problems. These skills are like tools in a mathematician's toolbox, ready to be used in different situations. So, by mastering this equation, we've not only found the value of x, but we've also strengthened our problem-solving abilities in mathematics.

Solving for variables in algebraic equations is a fundamental skill that opens the door to a world of mathematical possibilities. In this guide, we tackled the equation $9 - \frac{1}{4}x = \frac{1}{8}x$, and through a step-by-step approach, we not only found the solution but also learned the importance of verifying our answer. We started by clearing the fractions, simplifying the equation and making it easier to manipulate. Then, we isolated the variable x by combining like terms and performing algebraic operations on both sides of the equation. Finally, we arrived at the solution: $x = 24$. But we didn't stop there. We emphasized the crucial step of checking the solution by substituting it back into the original equation. This process confirmed that our answer was correct, giving us confidence in our solution. This skill of solving equations is not just about finding the right number; it's about developing a methodical approach to problem-solving. It's about understanding the principles of algebra and applying them strategically to unravel the unknown. The ability to solve for variables is essential in many fields, including science, engineering, economics, and computer science. It's a skill that empowers us to make predictions, solve problems, and understand the world around us. The importance of practice cannot be overstated. The more we practice solving equations, the more comfortable and confident we become. Each equation we solve is like a puzzle piece that fits into the larger picture of our mathematical understanding. So, keep practicing, keep exploring, and keep challenging yourself with new equations. Remember, guys, that mastering algebra is a journey, not a destination. It's a process of continuous learning and growth. The skills we acquire along the way will serve us well in our academic pursuits and beyond. So, embrace the challenge, enjoy the process, and celebrate your successes. You've got this!

$x = 24$