Solving The Equation: (u+6)/(u-1) + (u-4)/(u+1) = 2

by Admin 52 views
Solving the Equation: (u+6)/(u-1) + (u-4)/(u+1) = 2

Hey guys! Let's dive into solving this cool algebra problem: (u+6)/(u-1) + (u-4)/(u+1) = 2. It might look a bit intimidating at first, with those fractions and all, but trust me, we'll break it down step by step and make it super easy to understand. We'll be using some basic algebraic techniques to simplify things and isolate the variable 'u' to find its value. So, grab your pencils and let's get started on this math adventure! This problem is a classic example of how to handle equations involving rational expressions, and it's a fundamental concept in algebra. Being able to solve such problems is a key skill. Let's start with identifying the main goal: find the value(s) of 'u' that satisfy the equation. This involves a series of algebraic manipulations to eventually isolate 'u' on one side of the equation. We’ll be focusing on a clear, organized approach so you won’t get lost along the way. First things first, we need to deal with those pesky fractions. Our main strategy here will be to eliminate the fractions by multiplying both sides of the equation by a common denominator. This process will transform our equation into a more manageable form, which is crucial for solving it efficiently. Once we’ve eliminated the fractions, we'll have a polynomial equation that we can easily solve. Get ready, as this is a fundamental concept in algebra.

Step-by-Step Solution

  1. Finding the Common Denominator. Okay, so the denominators we're dealing with are (u-1) and (u+1). To get rid of the fractions, we need a common denominator. In this case, it's simply the product of the two denominators: (u-1)(u+1). Remember that a common denominator is the smallest expression that is a multiple of all the denominators in the equation. This will be the key to our next step, which is to eliminate the fractions, and is really cool, right? We multiply the entire equation by this common denominator. This ensures that when we simplify, the fractions will disappear, leaving us with a much simpler equation to deal with. This process is like a fundamental trick in algebra, and it makes things a whole lot easier. When you have multiple terms in your equation, multiplying each term by the common denominator can look a bit daunting, but stick with it, it's gonna work great!

  2. Multiplying by the Common Denominator. Let's multiply both sides of the equation by (u-1)(u+1):

[(u+6)/(u-1) + (u-4)/(u+1)] * (u-1)(u+1) = 2 * (u-1)(u+1)

Now, distribute (u-1)(u+1) to each term on the left side of the equation:

[(u+6)/(u-1)] * (u-1)(u+1) + [(u-4)/(u+1)] * (u-1)(u+1) = 2 * (u-1)(u+1)

Notice how (u-1) cancels out in the first term, and (u+1) cancels out in the second term. This is exactly what we wanted! Simplifying each term and expanding the right side:

(u+6)(u+1) + (u-4)(u-1) = 2(u^2 - 1)

Pretty cool, right? This step is all about making the equation easier to work with by getting rid of those fractions. It's a game changer when simplifying these types of algebraic problems. Remember to be careful and distribute correctly; it can be easy to make a small mistake here, so take your time and double-check your work!

  1. Expanding and Simplifying. Now, let's expand the terms and simplify the equation:

(u^2 + 7u + 6) + (u^2 - 5u + 4) = 2u^2 - 2

Combine like terms on the left side:

2u^2 + 2u + 10 = 2u^2 - 2

At this stage we are getting closer to our final solution. Now we combine like terms. This means we'll add and subtract terms with the same variable and exponent, and also combine any constants. Simplifying the equations is often the most important step in algebra. Always take your time and be careful. Double-check that you've correctly identified the like terms before combining them. Doing this ensures that you don't mess up your calculations. The more accurate you are here, the better your chances are of reaching the correct final answer!

  1. Isolating 'u'. To solve for 'u', we need to isolate it. Subtract 2u^2 from both sides of the equation:

2u + 10 = -2

Subtract 10 from both sides:

2u = -12

Divide both sides by 2:

u = -6

And there we have it! We've found the solution for 'u'. This is the core of solving the equation. Remember, our goal was to get 'u' all by itself on one side of the equation. This is the main aim for this algebra technique, and now we're almost there! Always double-check your calculations, especially during the isolating phase. Getting the correct final answer is a great feeling, and it builds your confidence in tackling similar problems in the future.

Checking the Solution

It's always a good idea to check your solution by plugging it back into the original equation to make sure it's correct. Let's substitute u = -6 into the original equation:

(-6+6)/(-6-1) + (-6-4)/(-6+1) = 2

0/(-7) + (-10)/(-5) = 2

0 + 2 = 2

2 = 2

Awesome! The solution checks out. This step is super important, especially when you are working on exams or tests. It confirms that the answer you've found is correct and that you haven't made any mistakes along the way. This also helps you understand the process and builds confidence. If your answer does not satisfy the original equation, then you should revisit each step carefully and look for any mistakes you might have made. It might be simple, such as a calculation error or a misplaced sign, so be patient and thorough! Checking your solution is not just about making sure you got the correct answer; it also helps reinforce your understanding and sharpens your skills!

Conclusion

Congratulations, guys! We've successfully solved the equation (u+6)/(u-1) + (u-4)/(u+1) = 2, and found that u = -6. This problem demonstrates how to handle rational equations, including the steps of finding a common denominator, eliminating fractions, and isolating the variable. Remember, the key to solving such problems is to break them down into smaller, manageable steps. This not only makes the process less overwhelming but also helps in avoiding errors. The skills you've acquired here are fundamental in algebra and will be useful in more advanced math topics. Keep practicing and exploring different types of algebraic problems to improve your problem-solving skills! Being comfortable with these skills will make learning advanced topics in math much easier. Always remember to check your solutions and keep practicing. The more you work with these types of problems, the easier and more natural it will become for you. Keep up the great work, and happy solving!