Solving For X: A Step-by-Step Guide
Hey there, math enthusiasts! Today, we're going to dive into the world of algebraic equations and solve for the value of x in the equation: . Don't worry, it might look a bit intimidating at first, but with a few simple steps, we'll break it down and find the solution. This is a classic example of a rational equation, and the key to solving these types of problems is to eliminate those pesky fractions. We will walk through the process, making sure you grasp every step. Grab your pencils and paper; let's get started!
Understanding the Problem: The Equation Unveiled
Understanding the equation is the first step toward solving it. The core of our problem is the equation . It's a rational equation, which means it involves fractions with variables in the denominator. Our mission is clear: isolate x and find its value(s) that make this equation true. Before we begin, it's essential to understand that there may be values of x that are not permissible. Specifically, values of x that make the denominator of any fraction equal to zero are excluded. In our equation, this means x cannot equal 1 or 0, as these values would result in division by zero, which is undefined. Keeping this in mind, let's go deeper into the method.
Identifying Potential Pitfalls
Before we jump in, let's talk about potential pitfalls. Rational equations often come with restrictions. In our equation, we have two denominators: x-1 and x. We need to identify any values of x that would make these denominators equal zero. If x-1 = 0, then x = 1. If x = 0, then, well, x = 0. So, we know that x cannot be 0 or 1. These are our restrictions, and any solutions we find must not be equal to these values. This step is critical; otherwise, we might find solutions that are not valid. Always watch out for these restrictions; they are our safety nets in the realm of rational equations. Now, with a clear understanding of the problem and our constraints, we are all set to find the solution. Let's move to the next stage where we will start simplifying the equation.
Why Restrictions Matter
You might be wondering why we bother with these restrictions. Well, imagine trying to divide something by zero. It's impossible, right? In the world of mathematics, it leads to undefined results, which are not considered valid solutions. By identifying restrictions early on, we prevent ourselves from accepting incorrect answers. For example, if we got x = 1 as a solution, we'd know immediately that something went wrong. So, restrictions are essential to ensure the solutions we arrive at are valid and make sense within the context of the equation. Always, always check for these potential problem areas before diving into the core of the problem. That's a golden rule for solving rational equations.
The Elimination Strategy: Clearing the Fractions
Okay, guys, it is time to start solving the equation. The best strategy is to get rid of those fractions. The initial equation is . To eliminate the fractions, we need to find the least common denominator (LCD) of all the fractions involved. In our case, the denominators are x-1 and x. Therefore, the LCD is simply x(x-1). We multiply both sides of the equation by this LCD. This is a crucial step in simplifying the equation.
Multiplying by the LCD: A Step-by-Step Guide
Let us go through the multiplication step-by-step. First, let us multiply each term by the LCD, which is x(x-1). Doing this carefully and systematically will ensure that we avoid any errors. Here is how it looks: x(x-1)[-\frac{3}{x-1}] - x(x-1)[\frac{1}{x}] = x(x-1)[-\frac{3x}{x-1}]. Now, simplify each term separately. For the first term, the (x-1) in the numerator and denominator cancel out, leaving us with -3x. For the second term, the x in the numerator and denominator cancel out, leaving us with -(x-1). And for the third term, the (x-1) cancels out, resulting in -3x². By multiplying each term by the LCD and carefully simplifying, the fractions magically disappear. The equation becomes much easier to deal with, allowing us to proceed towards finding the value of x.
Simplifying the Equation
After multiplying by the LCD and simplifying, our equation is now -3x - (x-1) = -3x². Further simplification is required to get a quadratic equation in standard form. First, distribute the negative sign in the second term: -3x - x + 1 = -3x². Then, combine like terms on the left side: -4x + 1 = -3x². Now, move all the terms to one side to get a standard quadratic equation: 3x² - 4x + 1 = 0. This step is crucial because it transforms our equation into a familiar format that we can solve using various methods, such as factoring or the quadratic formula. With this simplified and structured equation, the solution is just a step away.
Solving the Quadratic Equation: Finding the Roots
Now that we've cleared the fractions and simplified the equation into a standard quadratic form, it's time to solve for x. The equation is 3x² - 4x + 1 = 0. We can solve this equation in multiple ways: factoring, completing the square, or using the quadratic formula. Let us use factoring for this example. We are looking for two numbers that multiply to give 3 and add up to -4. The numbers are -3 and -1. The equation becomes (3x - 1)(x - 1) = 0. Now we can use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero.
Applying the Zero-Product Property
We will use the zero-product property to find the values of x. Setting each factor equal to zero: 3x - 1 = 0 and x - 1 = 0. Solving the first equation: 3x = 1, which gives us x = 1/3. Solving the second equation: x = 1. We have found two potential solutions: x = 1/3 and x = 1. But remember our restrictions? We established that x cannot equal 1 because it makes the denominator zero in the original equation. Therefore, we must check if our solutions align with these restrictions.
Checking the Solutions and Final Answer
We got two potential solutions, x = 1/3 and x = 1. But we know from earlier that x cannot be 1. It makes the denominator in our original equation equal to zero. Therefore, we reject x = 1 as an extraneous solution. An extraneous solution is a solution that arises from the process of solving the equation but does not satisfy the original equation. Now, let us check x = 1/3. Substituting x = 1/3 into the original equation, we do not get any division by zero and the equation holds true. This means x = 1/3 is a valid solution. Hence, the final answer is x = 1/3. We have successfully solved for x in the given rational equation. Hooray, you did it!
Conclusion: Wrapping It Up
Alright, folks, we've reached the finish line! Solving rational equations might seem tricky at first, but we've seen that it's all about systematically clearing fractions, simplifying, and checking for extraneous solutions. Always remember to identify restrictions early on to avoid accepting invalid answers. Congratulations on successfully solving this problem. You are now well-equipped to tackle similar challenges in the future! Keep practicing, and you'll become a pro at solving equations in no time! Keep learning, keep exploring, and until next time, happy calculating, guys!