Solving For U: (u-2)/(8u) - 1/12 = 1/u - Step-by-Step Guide
Hey guys! Today, we're going to tackle a fun little algebraic equation. We're aiming to solve for 'u' in the equation: (u-2)/(8u) - 1/12 = 1/u. Don't worry if it looks a bit intimidating at first glance. We'll break it down step-by-step so it's super easy to follow. So grab your pencils, paper, and let's dive right in!
1. Understanding the Equation and Identifying the Goal
Before we jump into solving, let's make sure we understand what we're looking at. Our equation is (u-2)/(8u) - 1/12 = 1/u. This is a rational equation, which simply means it involves fractions with variables in the denominator. Our main goal is to isolate 'u' on one side of the equation. To do this, we'll need to get rid of those pesky fractions. The best way to approach these types of problems is by finding the least common denominator (LCD) of all the fractions in the equation. This critical first step sets the stage for simplifying and ultimately solving for our variable, 'u'. Ignoring this step can lead to more complicated calculations and a higher chance of making errors. So, let's put on our thinking caps and figure out how to tackle those denominators!
Understanding the equation is paramount. We have fractions, and to effectively deal with them, we need a common denominator. This understanding guides our entire approach. The equation (u-2)/(8u) - 1/12 = 1/u involves three terms, each a fraction. The denominators are 8u, 12, and u. Our primary aim is to find a value for 'u' that makes this equation true. Isolating 'u' will involve clearing these fractions, which brings us to the concept of the least common denominator (LCD). The LCD is the smallest multiple that all denominators can divide into evenly. This is a crucial concept because multiplying each term in the equation by the LCD will eliminate the fractions, making the equation much easier to manipulate and solve. Think of it as finding the perfect key to unlock the solution. Without the LCD, we'd be stuck with complex fractions, and the process would be significantly harder. So, identifying and calculating the LCD is not just a step; it's the cornerstone of solving rational equations like this one. Remember, the goal is not just to find an answer, but to understand the process. This understanding will allow you to tackle similar problems with confidence. So, let's break down the denominators and find that LCD!. Using mathematical principles to your advantage is key here.
Remember, guys, solving for 'u' isn't just about finding a number; it's about understanding the relationship between the different parts of the equation. Each step we take brings us closer to that understanding, so let's keep going!
2. Finding the Least Common Denominator (LCD)
Okay, now for the fun part: finding the LCD. Look at our denominators: 8u, 12, and u. To find the LCD, we need to consider the prime factorization of the numbers and the variables involved. Let's break it down:
- 8u can be factored as 2 * 2 * 2 * u
- 12 can be factored as 2 * 2 * 3
- u is simply u
To build the LCD, we take the highest power of each factor that appears in any of the denominators. So, we need 2 * 2 * 2 (from the 8), 3 (from the 12), and u (from both 8u and u). Multiplying these together gives us 2 * 2 * 2 * 3 * u = 24u. So, our LCD is 24u. This is a crucial step, because it allows us to clear the fractions and simplify the equation.
Finding the LCD might seem like a small step, but it is absolutely critical to solving the equation efficiently. Without the LCD, we would be stuck dealing with complex fractions, making the algebra much more difficult. Now that we have the LCD, we can move on to the next step, which is multiplying both sides of the equation by 24u. This will eliminate the denominators and give us a much simpler equation to work with. Think of the LCD as the common ground that all the fractions can stand on, allowing us to combine and manipulate them easily. So, remember, taking the time to find the LCD correctly is an investment that pays off in the long run. It's like building a solid foundation for a house – you can't have a strong house without a strong foundation. Similarly, you can't solve a rational equation without a solid understanding of the LCD. We will use the properties of mathematical equations to our advantage in clearing out denominators, which is a powerful technique in algebra.
3. Multiplying Both Sides of the Equation by the LCD
Now that we've found our LCD (24u), we're going to multiply both sides of the equation by it. This is a key step to eliminate the fractions. Our equation is (u-2)/(8u) - 1/12 = 1/u. Let's multiply both sides by 24u:
24u * [(u-2)/(8u) - 1/12] = 24u * (1/u)
We need to distribute 24u on the left side:
24u * (u-2)/(8u) - 24u * (1/12) = 24u * (1/u)
Now, let's simplify each term:
- (24u * (u-2))/(8u) simplifies to 3(u-2)
- (24u * (1/12)) simplifies to 2u
- (24u * (1/u)) simplifies to 24
So our equation now looks like this:
3(u-2) - 2u = 24
See how much simpler it looks already? This step is crucial because it transforms a complex rational equation into a much easier linear equation. By eliminating the fractions, we've cleared the biggest hurdle and can now focus on solving for 'u' using basic algebraic techniques.
Multiplying both sides of the equation by the LCD is a powerful technique that allows us to get rid of the fractions. This simplification is essential for making the equation manageable. The concept behind this step is rooted in the fundamental principle that you can perform the same operation on both sides of an equation without changing its balance. It’s like a seesaw – if you add the same weight to both sides, it remains balanced. In our case, multiplying by 24u ensures that the equality remains valid while eliminating the denominators. Distributing the LCD across the terms on the left side might seem like a small detail, but it's vital for correct simplification. Each term needs to be multiplied individually to maintain the equation's integrity. The resulting equation, 3(u-2) - 2u = 24, is a testament to the power of this technique. It's a linear equation, which is much simpler to solve compared to the original rational equation. This step effectively bridges the gap between a complex problem and a straightforward solution. Remember, every simplification brings us closer to isolating 'u' and finding its value. Let us apply our knowledge of algebraic manipulation to arrive at a simpler equation.
4. Distributing and Simplifying the Equation
Time to distribute and simplify! We've got 3(u-2) - 2u = 24. First, let's distribute the 3:
3 * u - 3 * 2 - 2u = 24
This gives us:
3u - 6 - 2u = 24
Now, let's combine like terms (3u and -2u):
u - 6 = 24
See how much cleaner it's getting? Combining like terms is a fundamental step in simplifying algebraic equations. It reduces the number of terms and makes the equation easier to solve. By carefully applying the distributive property and combining like terms, we've transformed the equation into a more manageable form. This step is like decluttering a room – by organizing and simplifying, we make it easier to see what we have and where we need to go.
Distributing and simplifying is where we start to really see the equation transform. The distributive property allows us to multiply the 3 across the parentheses, eliminating them and giving us individual terms. This is a crucial step because it allows us to combine like terms, which further simplifies the equation. Think of distribution as expanding the equation, and combining like terms as condensing it. Together, these operations bring clarity and order to the equation, making it much easier to visualize the next steps. It’s like taking a complex map and highlighting the most important routes – we’re focusing on the essential elements that will lead us to the solution. The simplified equation, u - 6 = 24, is a testament to the power of these algebraic techniques. We are now on the home stretch and we are using our mastery of basic algebraic operations to clarify the equation.
5. Isolating 'u' and Solving for Its Value
Okay, we're in the home stretch! Our equation is u - 6 = 24. To isolate 'u', we need to get rid of the -6 on the left side. We can do this by adding 6 to both sides of the equation:
u - 6 + 6 = 24 + 6
This simplifies to:
u = 30
Yay! We've solved for u! u = 30. This is the value of 'u' that makes our original equation true. We've successfully isolated 'u' by performing the inverse operation (addition) to undo the subtraction. This is a core principle in algebra – to isolate a variable, we use inverse operations to cancel out terms on one side of the equation.
Isolating 'u' is the ultimate goal of solving the equation. It’s like reaching the summit of a mountain after a long climb. Adding 6 to both sides is the key move here, as it cancels out the -6 on the left side, leaving 'u' all by itself. This is based on the fundamental principle that performing the same operation on both sides of an equation maintains its balance. Think of it as adjusting a scale – if you add the same amount to both sides, the scale remains balanced. The result, u = 30, is our solution. This is the value that, when substituted back into the original equation, will make it true. It’s the culmination of all our efforts, a testament to the power of algebraic manipulation. However, it's not just about finding the answer; it's also about understanding the process. Knowing how to isolate a variable is a fundamental skill in algebra, and it’s a skill that will serve you well in many different contexts. We are employing the inverse operations strategy to unravel the equation step by step.
6. Checking the Solution
It's always a good idea to check our solution to make sure it's correct. To do this, we'll substitute u = 30 back into our original equation: (u-2)/(8u) - 1/12 = 1/u
Let's plug in u = 30:
(30-2)/(8*30) - 1/12 = 1/30
Simplify:
28/240 - 1/12 = 1/30
Reduce 28/240 to 7/60:
7/60 - 1/12 = 1/30
Find a common denominator (60) for the left side:
7/60 - 5/60 = 1/30
Simplify:
2/60 = 1/30
Reduce 2/60 to 1/30:
1/30 = 1/30
The equation holds true! So, our solution u = 30 is correct. Checking our solution is vital because it ensures that we haven't made any mistakes along the way. It's like proofreading a paper before submitting it – it helps catch any errors and ensures that our final answer is accurate. This step reinforces our confidence in the solution and demonstrates a thorough understanding of the problem-solving process.
Checking the solution is like the final exam – it's our opportunity to verify that all our work has been correct. Substituting u = 30 back into the original equation allows us to see if both sides of the equation are truly equal. This process involves careful simplification and arithmetic, and it's a crucial step in ensuring the accuracy of our answer. If the equation holds true, as it does in our case, we can be confident that our solution is correct. However, if the equation doesn't balance, it indicates that we've made an error somewhere in our calculations, and we need to go back and review our steps. Checking the solution is not just about getting the right answer; it's about developing good problem-solving habits and building confidence in our mathematical abilities. We apply the validation techniques to confirm the correctness of our solution.
Conclusion
And there you have it! We've successfully solved the equation (u-2)/(8u) - 1/12 = 1/u for u, and we found that u = 30. Remember, solving these types of equations is all about breaking them down into smaller, manageable steps. By finding the LCD, multiplying both sides, simplifying, and isolating the variable, we can tackle even the most intimidating-looking equations. Keep practicing, and you'll become a pro in no time!
So guys, remember, math might seem tricky sometimes, but with a step-by-step approach and a little bit of practice, you can conquer anything! Keep up the awesome work! This is our opportunity to summarize our journey through the problem-solving process.