Solving Rational Equations A Step By Step Guide

Introduction

Hey guys! Today, we're tackling a really interesting equation that involves fractions and variables. These types of equations might seem a little intimidating at first, but trust me, once you break them down step by step, they're totally manageable. We're going to solve the equation:

$\frac{6}{x-3}-\frac{6}{x}=\frac{18}{x^2-3 x}$

This equation falls under the category of rational equations, which basically means equations that contain fractions with variables in the denominator. Our goal is to find the value(s) of 'x' that make this equation true. So, grab your thinking caps, and let's dive in! This equation-solving journey will not only help you understand the specific steps involved but also enhance your overall problem-solving skills in mathematics. Understanding rational equations is crucial, as they appear in various fields, from physics and engineering to economics and computer science. So, by mastering these techniques, you're not just solving a math problem; you're also equipping yourself with valuable tools for future academic and professional endeavors. Let’s embark on this mathematical adventure together and unravel the mystery behind this equation! Remember, the key to success in mathematics is consistent practice and a clear understanding of the underlying concepts. As we go through each step, make sure to pause and reflect on why we're doing what we're doing. This will help you internalize the process and make you a more confident problem-solver. So, let's get started and conquer this equation!

Step 1: Identify the Domain

Before we even start manipulating the equation, it's super important to figure out the domain. What's the domain, you ask? Well, it's simply the set of all possible values of 'x' that we're allowed to plug into the equation without causing any mathematical mayhem. In other words, we need to identify any values of 'x' that would make the denominators of our fractions equal to zero, because dividing by zero is a big no-no in math. Looking at our equation:

$\frac{6}{x-3}-\frac{6}{x}=\frac{18}{x^2-3 x}$

We have three denominators: x - 3, x, and x^2 - 3x. Let's see what values of 'x' make these zero. For the first denominator, x - 3, we set it equal to zero and solve:

x - 3 = 0

x = 3

So, x = 3 is a value we need to exclude. Next, for the denominator x, it's pretty straightforward:

x = 0

So, x = 0 is another value we need to exclude. Finally, let's tackle the denominator x^2 - 3x. We can factor this expression:

x^2 - 3x = x(x - 3)

Setting this equal to zero gives us:

x(x - 3) = 0

This means either x = 0 or x - 3 = 0, which we already found. So, our domain is all real numbers except x = 0 and x = 3. We need to keep this in mind because if we find solutions that are 0 or 3, we'll have to throw them out – they're not valid solutions. Identifying the domain at the beginning is a crucial step in solving rational equations. It ensures that we don't end up with solutions that are mathematically undefined. Think of it as setting the rules of the game before we start playing. By understanding the domain, we avoid potential pitfalls and ensure that our final answers are valid and meaningful. This step highlights the importance of paying attention to the details and being thorough in our approach to problem-solving. So, remember, always start by identifying the domain!

Step 2: Find the Least Common Denominator (LCD)

Alright, now that we know which values of 'x' are off-limits, it's time to get rid of those pesky fractions! To do this, we're going to find the least common denominator (LCD) of all the fractions in our equation. The LCD is the smallest expression that each of our denominators can divide into evenly. This will allow us to multiply both sides of the equation by the LCD, which will clear out the fractions and make our equation much easier to work with. Looking at our denominators again:

• x - 3
• x
• x^2 - 3x

We already factored x^2 - 3x into x(x - 3). This makes finding the LCD much easier. The LCD needs to include all the factors present in each denominator, with the highest power of each factor. In this case, we have the factors x and (x - 3). So, our LCD is simply:

LCD = x(x - 3)

That wasn't too bad, right? Finding the LCD is a fundamental step in solving rational equations. It's like finding a common language that all the fractions can understand. By multiplying both sides of the equation by the LCD, we're essentially translating the equation into a simpler form that we can easily solve. This step demonstrates the power of strategic simplification in mathematics. By identifying the LCD, we're not just getting rid of fractions; we're also making the equation more manageable and reducing the chances of making errors. So, remember, finding the LCD is a crucial step towards solving rational equations efficiently and accurately. Now that we have our LCD, we're ready to move on to the next step: clearing the fractions.

Step 3: Multiply Both Sides by the LCD

Okay, we've got our LCD – x(x - 3) – and we're ready to put it to work! The next step is to multiply both sides of our equation by the LCD. This is where the magic happens, and we get to eliminate those fractions. Remember, whatever we do to one side of the equation, we have to do to the other side to keep things balanced. So, let's multiply both sides of our equation by x(x - 3):

x(x - 3) * [\frac{6}{x-3}-\frac{6}{x}] = x(x - 3) * [\frac{18}{x^2-3 x}]

Now, we need to distribute the LCD on both sides. Let's start with the left side. We'll multiply x(x - 3) by each term inside the brackets:

x(x - 3) * \frac{6}{x-3} - x(x - 3) * \frac{6}{x} = x(x - 3) * \frac{18}{x^2-3 x}

Now comes the fun part – canceling out common factors! In the first term, (x - 3) cancels out, leaving us with 6x. In the second term, x cancels out, leaving us with 6(x - 3). And on the right side, x(x - 3) is exactly the same as x^2 - 3x, so they cancel out completely. This simplifies our equation to:

6x - 6(x - 3) = 18

See how much simpler that looks? By multiplying both sides by the LCD, we've transformed our rational equation into a linear equation, which is much easier to solve. This step is a prime example of how algebraic manipulation can simplify complex problems. By strategically choosing to multiply by the LCD, we've cleared the fractions and paved the way for a straightforward solution. This technique is not just useful for this particular equation; it's a fundamental skill in algebra that you'll use again and again. So, remember, multiplying by the LCD is a powerful tool for solving rational equations. Now that we've cleared the fractions, let's move on to the next step: simplifying and solving the resulting equation.

Step 4: Simplify and Solve the Equation

Alright, we've successfully cleared the fractions, and our equation is looking much cleaner: 6x - 6(x - 3) = 18. Now, it's time to simplify and solve for 'x'. First, let's distribute the -6 in the second term:

6x - 6x + 18 = 18

Notice anything interesting? The 6x and -6x terms cancel each other out! This leaves us with:

18 = 18

Whoa, what does this mean? We've ended up with a statement that's always true, regardless of the value of 'x'. This is a special case called an identity. An identity means that the equation is true for all values of 'x' in the domain. But hold on a second! Remember back in Step 1 when we identified our domain? We said that x cannot be 0 or 3 because those values would make our denominators zero. So, even though the equation 18 = 18 is always true, our original equation has restrictions on 'x'. This means that the solution set is all real numbers except 0 and 3. This step highlights the importance of considering the domain when solving equations. Even if we arrive at a seemingly straightforward result, we must always check it against the original restrictions. In this case, we found an identity, but the domain limitations prevent us from saying that all real numbers are solutions. This is a crucial lesson in mathematical problem-solving: always be mindful of the context and the initial conditions. So, what does this mean for our answer? Well, since the equation is an identity, it would be true for any value of x... if it weren't for those pesky restrictions on the domain we found in Step 1! This is a super important point: Always, always, always consider the domain when solving rational equations (or any equation with restrictions, for that matter). We found that x cannot be 0 or 3, so those values are out. So, while the equation itself simplifies to a true statement, the original equation doesn't have a solution because any value we plug in will either make a denominator zero (which is a big no-no) or is specifically excluded from our solution set. So, let's recap what we've learned in this step. We simplified the equation, and we found an identity. But we didn't stop there! We went back to our domain and realized that the restrictions on 'x' mean there's no solution to the original equation. This kind of attention to detail is what turns a good problem-solver into a great problem-solver.

Step 5: Check for Extraneous Solutions

Okay, so we've simplified our equation, and it seems like we've hit a bit of a snag. We ended up with 18 = 18, which is an identity, but we also know that x cannot be 0 or 3. This is where we need to be extra careful and check for extraneous solutions. Extraneous solutions are values that we might get as solutions during the solving process, but they don't actually work in the original equation. They often arise when we perform operations that can introduce new solutions, like squaring both sides of an equation or, in our case, multiplying by an expression containing 'x'. In our situation, we didn't find any specific numerical solutions, but we did find that the equation simplifies to an identity, suggesting that any value of 'x' could work... if it weren't for the domain restrictions. So, we need to go back to our original equation and see if any of our restricted values (0 and 3) would cause problems. Let's think about what happens if we plug x = 0 into our original equation:

$\frac{6}{x-3}-\frac{6}{x}=\frac{18}{x^2-3 x}$

We'd have a division by zero in the second term (\frac{6}{0}), which is undefined. So, x = 0 is definitely an extraneous solution. Now, let's try x = 3:

We'd have a division by zero in the first term (\frac{6}{3-3} = \frac{6}{0}) and also in the denominator on the right side (3^2 - 3*3 = 0). So, x = 3 is also an extraneous solution. Since both of our restricted values are extraneous, and we didn't find any other potential solutions, this means our original equation has no solution. This might seem a little disappointing, but it's a perfectly valid outcome in mathematics. Not every equation has a solution, and it's important to be able to recognize when that's the case. Checking for extraneous solutions is a crucial step in solving rational equations. It's like double-checking your work to make sure you haven't made any mistakes. By plugging our potential solutions back into the original equation, we can catch any extraneous solutions and ensure that our final answer is correct. So, remember, always check for extraneous solutions! This step reinforces the importance of being thorough and meticulous in our problem-solving approach. It's not enough to just find potential solutions; we must also verify that they actually work in the original context of the problem. This attention to detail is what distinguishes a good problem-solver from an excellent one.

Conclusion

So, there you have it! We've tackled a rational equation step by step, and we've discovered that it has no solution. This might not have been the outcome we initially expected, but it's a valuable lesson in mathematical problem-solving. We learned how to identify the domain, find the LCD, clear fractions, simplify equations, and, most importantly, check for extraneous solutions. These are skills that will serve you well in all your mathematical endeavors. Remember, math isn't just about finding the right answer; it's about the process of getting there. By understanding the steps involved and why they're necessary, you're building a strong foundation for future success. So, keep practicing, keep exploring, and keep challenging yourself! And remember, even when an equation has no solution, the journey of trying to solve it is still a worthwhile learning experience. We've reinforced the importance of domain restrictions and how they can impact the final solution. We've also highlighted the significance of checking for extraneous solutions, which can sometimes lead us astray if we're not careful. So, the next time you encounter a rational equation, remember these steps, and you'll be well-equipped to tackle it head-on. Keep up the great work, and happy problem-solving!

Correct Choice: The solution(s) is/are $\boxed{no solution}$.