Solving Rational Equations: A Step-by-Step Guide

Hey guys! Today, we're going to dive deep into the world of rational equations and tackle a common problem: solving for the unknown variable. Specifically, we'll be breaking down the equation 8/(p-5) = -6/(3p+1). If you've ever felt intimidated by fractions and variables mixed together, don't worry! We'll take it one step at a time, making sure you understand the logic behind each move. So, grab your pencils and notebooks, and let's get started!

Understanding Rational Equations

Before we jump into the solution, let's quickly recap what rational equations are all about. In essence, a rational equation is any equation that contains one or more rational expressions. A rational expression, in turn, is simply a fraction where the numerator and/or the denominator are polynomials. Think of it as a fancy way of saying equations with fractions where variables might be lurking in the denominator. For example, the equation we're tackling today, 8/(p-5) = -6/(3p+1), perfectly fits this definition. We have fractions, and the denominators contain our variable, p. Understanding this fundamental concept is crucial because it dictates the strategies we'll use to solve these equations. We need to be mindful of those denominators and make sure they don't cause any mathematical mayhem, like dividing by zero, which is a big no-no in the math world. So, keep in mind that dealing with rational equations is all about carefully maneuvering the fractions to isolate our variable, all while ensuring we're playing by the rules of algebra.

Step 1: Identifying Restrictions

Okay, guys, before we start crunching numbers, there's a crucial step we need to take: identifying restrictions. Think of restrictions as the 'no-go zones' for our variable. In the context of rational equations, these are the values of the variable that would make the denominator of any fraction equal to zero. Why is this so important? Because division by zero is undefined in mathematics – it's a mathematical black hole! So, we need to figure out which values of p would cause this to happen in our equation, 8/(p-5) = -6/(3p+1). Let's look at each denominator separately. First, we have (p-5). To find the restriction, we set it equal to zero: p - 5 = 0. Solving for p, we get p = 5. This means that p cannot be 5, because if it were, the denominator would be zero, and the equation would be undefined. Next, we have (3p+1). Again, we set it equal to zero: 3p + 1 = 0. Solving for p, we first subtract 1 from both sides: 3p = -1. Then, we divide both sides by 3: p = -1/3. So, p also cannot be -1/3. Therefore, our restrictions are p ≠ 5 and p ≠ -1/3. Make a note of these! They are our boundaries. If we get either of these values as a solution at the end, we know we've made a mistake somewhere, or the solution is extraneous. Identifying these restrictions upfront is like setting up safety parameters before embarking on a mathematical journey. It helps us avoid potential pitfalls and ensures that our final answer is valid and meaningful.

Step 2: Eliminating the Denominators

Now, let's get to the fun part: eliminating the denominators! This is the key step in solving rational equations because it transforms our equation from a potentially scary fraction-filled problem into a much friendlier linear equation. The trick here is to multiply both sides of the equation by the least common denominator (LCD). Think of the LCD as the smallest expression that each of our denominators can divide into evenly. In our equation, 8/(p-5) = -6/(3p+1), the denominators are (p-5) and (3p+1). Since these expressions don't share any common factors, their LCD is simply their product: (p-5)(3p+1). Now, we multiply both sides of the equation by this LCD. On the left side, (p-5) in the LCD cancels out with the (p-5) in the denominator, leaving us with 8(3p+1). On the right side, (3p+1) in the LCD cancels out with the (3p+1) in the denominator, leaving us with -6(p-5). So, our equation now looks like this: 8(3p+1) = -6(p-5). See how the fractions are gone? We've successfully eliminated the denominators! This step is so powerful because it simplifies the equation significantly, making it much easier to solve. By multiplying by the LCD, we're essentially 'clearing out' the fractions, paving the way for us to isolate our variable and find its value. This technique is a cornerstone of solving rational equations, and mastering it will make you a pro at handling these types of problems.

Step 3: Simplifying and Solving the Equation

Alright, with the denominators out of the picture, we're now ready to simplify and solve the equation. Our equation currently looks like this: 8(3p+1) = -6(p-5). The first thing we need to do is distribute the numbers outside the parentheses. On the left side, we distribute the 8: 8 * 3p = 24p and 8 * 1 = 8. So, the left side becomes 24p + 8. On the right side, we distribute the -6: -6 * p = -6p and -6 * -5 = 30. So, the right side becomes -6p + 30. Now our equation looks like this: 24p + 8 = -6p + 30. It's looking much simpler already! Next, we want to get all the p terms on one side of the equation and all the constant terms on the other side. Let's add 6p to both sides: 24p + 6p + 8 = -6p + 6p + 30, which simplifies to 30p + 8 = 30. Now, let's subtract 8 from both sides: 30p + 8 - 8 = 30 - 8, which simplifies to 30p = 22. Finally, to isolate p, we divide both sides by 30: (30p)/30 = 22/30. This gives us p = 22/30. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So, 22/30 simplifies to 11/15. Therefore, our solution is p = 11/15. Simplifying and solving the equation involves a series of algebraic manipulations, but by following the order of operations and carefully applying the properties of equality, we can systematically isolate our variable and find its value. This step is where all our hard work pays off, bringing us closer to the final answer.

Step 4: Checking for Extraneous Solutions

We've found a potential solution, but we're not done yet! The final, and arguably most important, step is checking for extraneous solutions. Remember those restrictions we identified way back in Step 1? This is where they come into play. Extraneous solutions are solutions that we obtain through the algebraic process, but they don't actually satisfy the original equation. They usually arise in rational equations (and radical equations) because of the way we manipulate the equation – specifically, when we eliminate denominators by multiplying both sides by an expression that contains a variable. Our solution is p = 11/15, and our restrictions were p ≠ 5 and p ≠ -1/3. Since 11/15 is not equal to 5 or -1/3, it doesn't violate our restrictions. However, we still need to plug p = 11/15 back into the original equation, 8/(p-5) = -6/(3p+1), to make sure it actually works. Let's substitute p = 11/15 into the left side: 8/((11/15)-5). To subtract 5 from 11/15, we need to convert 5 to a fraction with a denominator of 15: 5 = 75/15. So, (11/15) - (75/15) = -64/15. Therefore, the left side becomes 8/(-64/15). To divide by a fraction, we multiply by its reciprocal: 8 * (-15/64) = -120/64, which simplifies to -15/8. Now, let's substitute p = 11/15 into the right side: -6/(3(11/15)+1). First, we multiply 3 by 11/15: 3 * (11/15) = 33/15. Now, we add 1, which we need to convert to a fraction with a denominator of 15: 1 = 15/15. So, (33/15) + (15/15) = 48/15. Therefore, the right side becomes -6/(48/15). Again, we multiply by the reciprocal: -6 * (15/48) = -90/48, which simplifies to -15/8. Hooray! The left side (-15/8) equals the right side (-15/8). This confirms that p = 11/15 is indeed a valid solution. Checking for extraneous solutions is like the final quality control check in our solving process. It ensures that our algebraic manipulations haven't led us astray and that our solution is a true solution to the original equation.

Conclusion

So, guys, we've successfully solved the rational equation 8/(p-5) = -6/(3p+1)! We walked through each step carefully, from identifying restrictions to eliminating denominators, simplifying the equation, and finally, checking for extraneous solutions. The solution we found is p = 11/15. Remember, the key to mastering rational equations is to understand the underlying concepts and to be meticulous in your algebraic manipulations. Don't be afraid of fractions – embrace them! With practice and a systematic approach, you'll be solving rational equations like a pro in no time. Keep practicing, and you'll become a math whiz in no time!