Solve For X: Step-by-Step Guide

Are you struggling with equations that involve fractions and variables? Don't worry, you're not alone! Many students find solving for x in equations like this a bit tricky at first. But with a clear understanding of the steps involved, you'll be able to tackle these problems with confidence. In this guide, we'll break down the equation (x-2)/x - 5/3 = 4/x, step-by-step, so you can not only solve it but also understand the underlying principles. So, let's get started, guys!

Understanding the Equation

Before we dive into the solution, let's take a moment to understand what the equation is telling us. We have an equation with x in the denominator, which means x cannot be zero (because division by zero is undefined). This is a crucial point to remember. Our goal is to isolate x on one side of the equation. To do this, we'll need to get rid of the fractions and simplify the expression. Think of it like untangling a knot – we'll carefully undo each step until we have x all by itself. The given equation is: (x-2)/x - 5/3 = 4/x. This equation involves fractions, so our primary goal is to eliminate these fractions to simplify the equation and make it easier to solve for x. We'll achieve this by finding a common denominator and multiplying both sides of the equation by it. Remember, what we do to one side of the equation, we must do to the other to maintain balance.

Identifying Key Components

The equation consists of several terms: (x-2)/x, -5/3, and 4/x. Notice that x appears in the denominator of two terms. This is important because it tells us that x cannot be equal to zero. Why? Because division by zero is undefined in mathematics. So, right off the bat, we know that x = 0 is not a possible solution. This constraint is crucial to keep in mind as we proceed with solving the equation. Overlooking such restrictions can lead to incorrect solutions. So, always double-check for any values that might make the denominator zero and exclude them from your possible solutions. Another key aspect of this equation is the presence of fractions. Dealing with fractions can sometimes be tricky, so our first step will be to eliminate them. We'll do this by finding a common denominator for all the fractions in the equation. Once we have a common denominator, we can multiply both sides of the equation by it, effectively clearing the fractions and simplifying the equation. This will make the equation much easier to work with and solve for x. This step is essential because it transforms the equation into a more manageable form, allowing us to apply basic algebraic techniques.

Why We Need to Eliminate Fractions

Why go through the trouble of eliminating fractions? Well, working with fractions can be cumbersome. Imagine trying to add, subtract, multiply, or divide fractions – it requires finding common denominators, and sometimes, the numbers can get messy. By eliminating fractions, we transform the equation into a simpler form, usually a linear equation or a polynomial equation, which are much easier to solve. Think of it like this: you're building a house, and fractions are like oddly shaped bricks. It's possible to build with them, but it's much easier if you have regular, uniform bricks. Eliminating fractions is like shaping those bricks into a standard size, making the construction process smoother. The primary reason for eliminating fractions is to simplify the equation. Equations involving fractions often require extra steps and careful attention to detail to avoid errors. By clearing the fractions, we reduce the chances of making mistakes and make the algebraic manipulations much more straightforward. This simplification allows us to focus on the core task of isolating x and finding its value. Moreover, eliminating fractions often leads to a more recognizable form of the equation, such as a linear equation (ax + b = 0) or a quadratic equation (ax^2 + bx + c = 0). These forms have well-established methods for solving them. So, by transforming our original equation into one of these forms, we can apply standard techniques to find the solution for x.

Step-by-Step Solution

Now, let's get down to the nitty-gritty and solve the equation. Remember, our equation is: (x-2)/x - 5/3 = 4/x. The first step, as we discussed, is to eliminate the fractions. To do this, we need to find the least common denominator (LCD) of all the denominators in the equation. In our case, the denominators are x and 3. The LCD is the smallest multiple that both x and 3 divide into evenly, which is 3x. Now, we'll multiply both sides of the equation by 3x. This is a crucial step because it clears the fractions, making the equation much easier to handle. When we multiply both sides by 3x, we're essentially multiplying each term in the equation by 3x. This ensures that the equation remains balanced – whatever we do to one side, we must do to the other. After this multiplication, we'll simplify the equation by canceling out common factors. This will result in an equation without fractions, which we can then solve using standard algebraic techniques. Let's dive into the detailed steps.

1. Find the Least Common Denominator (LCD)

The denominators in our equation are x and 3. To find the least common denominator (LCD), we need to find the smallest expression that both x and 3 divide into evenly. In this case, the LCD is simply the product of x and 3, which is 3x. Finding the LCD is a crucial step in eliminating fractions from an equation. The LCD serves as the common multiple that allows us to combine or eliminate fractional terms. Without finding the LCD, it would be difficult to perform operations such as addition or subtraction on fractions with different denominators. Think of it like trying to add apples and oranges – you need a common unit (like fruit) to make the addition meaningful. The LCD provides this common unit for our fractions. In more complex equations, finding the LCD might involve factoring the denominators and identifying the common and unique factors. For example, if the denominators were x^2 - 4 and x + 2, we would need to factor x^2 - 4 into (x + 2)(x - 2) to determine the LCD. However, in our case, the denominators are relatively simple, making the LCD straightforward to find. Once we have the LCD, we can proceed to the next step, which involves multiplying both sides of the equation by the LCD to eliminate the fractions. This step is the key to simplifying the equation and making it solvable.

2. Multiply Both Sides by the LCD

Now that we've found the LCD to be 3x, we'll multiply both sides of the equation by this value. This is the heart of the fraction-elimination process. Multiplying both sides of the equation (x-2)/x - 5/3 = 4/x by 3x gives us: 3x * [(x-2)/x - 5/3] = 3x * [4/x]. Remember, whatever we do to one side of the equation, we must do to the other to maintain the balance. It's like a seesaw – if you add weight to one side, you need to add the same weight to the other side to keep it level. Multiplying both sides by the LCD is a fundamental algebraic principle that ensures the equality of the equation is preserved. This step might seem a bit daunting at first, but it's actually quite straightforward. We're essentially distributing 3x to each term on both sides of the equation. This means we'll multiply 3x by (x-2)/x, then by -5/3, and finally by 4/x. Each of these multiplications will help us to cancel out the denominators and clear the fractions. The next step involves simplifying the resulting expression after the multiplication. We'll cancel out common factors in the numerators and denominators, which will lead us to an equation without fractions. This simplified equation will be much easier to solve for x. So, let's proceed with the distribution and simplification to see how the fractions disappear.

3. Simplify the Equation

After multiplying both sides by the LCD, 3x, we have: 3x * (x-2)/x - 3x * 5/3 = 3x * 4/x. Now, let's simplify each term by canceling out common factors. In the first term, 3x * (x-2)/x, the x in the numerator and denominator cancels out, leaving us with 3*(x-2). In the second term, 3x * 5/3, the 3 in the numerator and denominator cancels out, leaving us with 5x. In the third term, 3x * 4/x, the x in the numerator and denominator cancels out, leaving us with 34. So, our equation now becomes: 3(x-2) - 5x = 3*4. This equation looks much simpler than our original equation, doesn't it? We've successfully eliminated the fractions, which was our primary goal. The next step involves expanding the terms and combining like terms. We'll distribute the 3 in the first term and simplify the right side of the equation. This will bring us closer to isolating x and finding its value. Simplifying an equation is like tidying up a room – you want to group similar items together and remove any clutter. In our case, we're grouping the x terms and the constant terms together to make the equation easier to solve.

4. Distribute and Combine Like Terms

Let's continue simplifying our equation: 3*(x-2) - 5x = 34. First, we'll distribute the 3 in the first term: 3x* - 6 - 5x = 12. Now, we'll combine like terms. We have 3x and -5x on the left side, which combine to give -2x. So, our equation becomes: -2x - 6 = 12. The goal here is to isolate the term with x on one side of the equation. To do this, we'll add 6 to both sides of the equation. This will cancel out the -6 on the left side, leaving us with just the term with x. Adding the same value to both sides maintains the balance of the equation, ensuring that we're not changing the solution. Combining like terms is a crucial step in solving equations. It allows us to reduce the number of terms in the equation and simplify the expression. Without combining like terms, the equation can become unnecessarily complex and difficult to solve. Think of it like sorting your laundry – you group similar items together (like socks or shirts) to make it easier to manage. In the same way, we're grouping the x terms and the constant terms to make the equation easier to solve. The next step involves isolating x by performing the necessary operations. This will lead us to the final solution for x.

5. Isolate x

Our equation is now: -2x - 6 = 12. To isolate x, we need to get it by itself on one side of the equation. The first step is to add 6 to both sides: -2x - 6 + 6 = 12 + 6, which simplifies to -2x = 18. Now, we have -2x equal to 18. To solve for x, we need to divide both sides by -2: (-2x) / -2 = 18 / -2. This gives us x = -9. And there you have it! We've successfully solved for x. Isolating x is the final step in solving for the variable. It involves performing inverse operations to undo any operations that are being applied to x. For example, if x is being multiplied by a number, we divide both sides by that number. If a number is being added to x, we subtract that number from both sides. The goal is to get x alone on one side of the equation, so we can determine its value. After isolating x, it's always a good idea to check your solution by plugging it back into the original equation. This ensures that your solution is correct and satisfies the equation. We'll do this in the next step to verify our answer.

6. Check Your Solution

It's always a good practice to check your solution to make sure it's correct. We found that x = -9. Let's plug this value back into the original equation: (x-2)/x - 5/3 = 4/x. Substituting x = -9, we get: (-9-2)/(-9) - 5/3 = 4/(-9). Simplifying the left side: (-11)/(-9) - 5/3 = 11/9 - 5/3. To subtract these fractions, we need a common denominator, which is 9. So, we rewrite 5/3 as 15/9: 11/9 - 15/9 = -4/9. Now, let's look at the right side of the equation: 4/(-9) = -4/9. Since the left side (-4/9) is equal to the right side (-4/9), our solution x = -9 is correct! Checking your solution is like proofreading your work – it helps you catch any mistakes and ensures that your answer is accurate. By plugging the solution back into the original equation, we're verifying that it satisfies the equation and makes it true. This step is particularly important in equations involving fractions or radicals, where it's easy to make algebraic errors. If the left side of the equation equals the right side after substituting the solution, then you can be confident that your answer is correct. If the two sides are not equal, then you'll need to go back and review your steps to find the mistake. In our case, the solution checked out, so we can confidently say that x = -9 is the correct answer.

Conclusion

So, there you have it, guys! We've successfully solved for x in the equation (x-2)/x - 5/3 = 4/x. We found that x = -9. Remember, the key steps were: finding the LCD, multiplying both sides by the LCD, simplifying the equation, distributing and combining like terms, isolating x, and checking our solution. Solving equations like this can seem challenging at first, but with practice and a clear understanding of the steps, you'll become a pro in no time. Keep practicing, and don't be afraid to ask for help when you need it. Math is like a muscle – the more you exercise it, the stronger it gets. So, keep solving those equations, and you'll be amazed at what you can achieve! Remember, the most important thing is to understand the process, not just memorize the steps. When you understand why you're doing each step, you'll be able to apply these techniques to a wide variety of problems. So, keep exploring, keep learning, and most importantly, have fun with math!