Introduction: Unraveling the Formula
Hey guys! Ever stumbled upon a formula and thought, "Whoa, how do I even begin to solve this?" Well, let's break down a common problem in algebra: solving for a specific variable within an equation. Specifically, we're diving into the formula $G=\frac{9wu}{10}$. Our mission? To isolate w and express it in terms of G and u. Sounds intimidating? Don't worry, it's like a puzzle. We'll go step-by-step, and by the end, you'll be solving these problems like a pro. This guide is designed to be super clear, friendly, and easy to follow. We will make use of different mathematical concepts such as multiplication, division, and how to use parenthesis. So, buckle up, grab a pen and paper, and let's start this journey! The key to solving for any variable is to get it all alone on one side of the equation. We achieve this by performing inverse operations – operations that undo each other. For example, division undoes multiplication, and addition undoes subtraction. Our goal here is to isolate w, and we will go about doing this step by step. There are many things we can solve using this type of formula. First, we can find the weight given the parameters. Second, we can find the value of u and w under the desired parameters. We can also find the value of G when we provide the value of u and w. Therefore, understanding how to solve this equation is essential for various applications. Keep your eye on the goal: Getting w by itself. With each step, we'll get closer to the solution. Remember, in math, patience and a systematic approach are your best friends. Let's begin!
Step 1: Clearing the Fraction
Alright, let's tackle that fraction first. See that 10 in the denominator? It's currently dividing the entire right side of the equation. To get rid of it, we perform the inverse operation: multiplication. We're going to multiply both sides of the equation by 10. Why both sides? Because whatever we do to one side of an equation, we must do to the other to keep things balanced – this is a fundamental rule in algebra. Doing so ensures that our solution will be valid, no matter what. So, starting with our original equation, $G=\frac9wu}{10}$, we multiply both sides by 10{10}$. On the right side, the 10s cancel out, leaving us with $10G = 9wu$. Notice how the fraction is gone? That's the magic of inverse operations. Now, we've simplified our equation and made it a bit easier to work with. This step removes the immediate complexity of the fraction, setting the stage for isolating w more easily. Remember, with each step, we are closer to finding the correct answer. Keep up the hard work, and you are going to get there. Understanding this step is critical in many algebraic manipulations. You are on the right track!
Step 2: Isolating w
Okay, we've gotten rid of the fraction. Now, let's focus on getting w alone. Currently, w is being multiplied by both 9 and u. To isolate w, we need to undo those multiplications. The correct approach would be to perform the inverse operations. We're going to divide both sides of the equation by 9u. Remember, we do this to both sides to keep the equation balanced. So, our equation from the previous step was $10G = 9wu$. Now, we divide both sides by $9u$: $\frac10G}{9u} = \frac{9wu}{9u}$. On the right side, the 9s and the us cancel out, leaving us with just w. This isolation is a crucial step, bringing us closer to solving for w. On the left side, we have $\frac{10G}{9u}$. So, our equation now looks like this{9u} = w$. We have solved for w. We did it! w is all alone on one side, and we've expressed it in terms of G and u. Now you know the power of inverse operations and a step-by-step method. By following these steps, you will be able to deal with more complex algebraic equations in the future. Remember to stay focused and have fun while doing so. This step is critical to obtain the value of w. Do not get discouraged, you are almost there!
Step 3: The Solution: w Expressed
Congratulations, we have successfully isolated w! Our final answer is $w = \frac{10G}{9u}$. This is the solution to the equation $G=\frac{9wu}{10}$ when we solve for w. This equation tells us that w is equal to 10 times G, divided by 9 times u. Always double-check your work. Make sure your steps are logical and that you haven't made any arithmetic errors. The key here is to understand the process: How to manipulate equations using inverse operations to isolate a variable. Once you get the hang of it, solving these types of problems will become second nature. This is a super important concept in algebra, and now you have a solid understanding. You've not only solved for w but also gained valuable skills in algebraic manipulation. Keep practicing, and you will be able to solve more complex mathematical equations! Remember, mathematics is a skill that improves with practice. The more problems you solve, the better you become. So, do not be afraid to keep practicing; you are on the right track!
Step 4: Examples to Practice
Here are some example problems for you to try. These problems are similar to the one we just solved but with different numbers and formulas. Try to solve for the indicated variables and see if you can apply what you've learned:
- Example 1: Given $A = \frac{1}{2}bh$, solve for b.
- Example 2: Given $P = 2l + 2w$, solve for l.
- Example 3: Given $V = \frac{4}{3}\pi r^3$, solve for r.
These examples will help solidify your understanding and prepare you for more advanced algebraic concepts. Don't worry if you get stuck; the process is what matters. The more problems you attempt, the more comfortable you will become with these types of equations. Always remember to double-check your work and focus on the steps. By working through these examples, you will build confidence and proficiency in solving for variables in equations. Feel free to create your own examples, focusing on the key steps. This will allow you to solidify the concepts and make them stick. Remember, the key to mastering algebra is consistent practice and a willingness to learn. Keep up the great work, and you'll be solving more complex problems in no time! These example problems provide practical application of the concepts learned.
Conclusion: Mastering Algebraic Manipulation
Awesome job, guys! We've successfully navigated the process of solving for w in the formula $G=\frac{9wu}{10}$. You've learned how to clear fractions, use inverse operations, and isolate a variable. The ability to solve for variables is a fundamental skill in algebra, and now you've got a solid grasp of it. This is a skill that will be super useful in all sorts of mathematical and real-world situations. Remember the steps: clear fractions by multiplying by the denominator, isolate the variable by using inverse operations, and always keep the equation balanced by doing the same thing to both sides. Keep practicing, and you'll become a pro in no time! The more you practice, the more comfortable you will become with these types of problems. Always double-check your work and don't be afraid to ask for help if you get stuck. Learning mathematics is a journey, and every step you take, no matter how small, is a victory. Congratulation on solving the equation, I am super proud of you!