Solving For X In The Equation (x-7)/5 - (x-11)/6 A Step-by-Step Guide

Hey guys! Ever stumbled upon an equation that looks like a jumbled mess of fractions and variables? Don't worry, we've all been there. Today, we're going to break down one of those equations step-by-step, making it super easy to understand. We're diving into the world of algebra to find the value of 'x' in a fraction equation. So, buckle up and let's get started!

Understanding the Equation

Before we jump into solving, let's make sure we understand the equation we're dealing with. We have a fraction equation with 'x' in the numerator. Our mission is to isolate 'x' and figure out its numerical value. Think of it like a puzzle where 'x' is the missing piece. Our job is to find that missing piece by carefully manipulating the equation. It might seem daunting at first, but trust me, with a few simple steps, we can crack this code. We'll start by identifying the different parts of the equation, then we'll use some algebraic techniques to simplify things. Remember, the key is to keep the equation balanced – whatever we do to one side, we must do to the other. This ensures that the value of 'x' remains the same throughout our calculations. So, let's put on our detective hats and start our quest to find 'x'!

Step 1: Clearing the Fractions

One of the most common hurdles in solving fraction equations is, well, the fractions themselves! They can make things look complicated and intimidating. But don't fret! There's a simple trick to get rid of them: we can multiply both sides of the equation by the least common multiple (LCM) of the denominators. The least common multiple (LCM) is the smallest number that is a multiple of both denominators. This magical number allows us to eliminate the fractions, making the equation much easier to work with. So, first things first, we need to identify the denominators in our equation. Once we have those, we'll find their LCM. Then, we'll multiply both sides of the equation by this LCM, ensuring that we're maintaining the balance. This step is crucial because it transforms our complex fraction equation into a simpler, more manageable form. Think of it as clearing away the clutter so we can see the underlying structure of the equation more clearly. With the fractions gone, we'll be one step closer to finding the elusive 'x'.

Step 2: Simplifying the Equation

Now that we've cleared the fractions, it's time to simplify the equation further. This usually involves distributing any multiplication and combining like terms. Distribution is like handing out candy to everyone in the parenthesis – we multiply the number outside the parenthesis by each term inside. Once we've distributed, we'll have a bunch of terms, some with 'x' and some without. That's where combining like terms comes in. Think of it as sorting your socks – you put all the pairs together. Similarly, we'll group all the 'x' terms together and all the constant terms (the ones without 'x') together. This step is like tidying up the equation, making it more organized and easier to read. By simplifying, we're reducing the equation to its most basic form, which makes the next steps much smoother. A simplified equation is a happy equation, and it brings us closer to our goal of finding 'x'. So, let's roll up our sleeves and get simplifying!

Step 3: Isolating 'x'

Alright, we're getting closer to the finish line! Now comes the crucial step of isolating 'x'. This means getting 'x' all by itself on one side of the equation. To do this, we'll use inverse operations. Think of inverse operations as the undo buttons of math. If something is being added to 'x', we'll subtract it from both sides. If something is being multiplied by 'x', we'll divide both sides. Remember, the golden rule is to do the same thing to both sides of the equation to keep it balanced. It's like a seesaw – if you add weight to one side, you need to add the same weight to the other to keep it level. This step might involve a few moves, like adding or subtracting constants and then dividing by the coefficient of 'x' (the number multiplied by 'x'). But with each move, we're peeling away the layers surrounding 'x', gradually revealing its true value. Isolating 'x' is like freeing it from its mathematical prison, and once it's free, we'll have our answer!

Step 4: Solving for 'x'

We've reached the moment of truth – it's time to solve for 'x'! After all the simplifying and isolating, we should now have an equation that looks something like 'x = some number'. That 'some number' is the value of 'x' that makes the original equation true. But before we declare victory, it's always a good idea to double-check our work. We can do this by plugging our value of 'x' back into the original equation and seeing if both sides are equal. This is like checking your answer in a test – it gives you the confidence that you've got it right. If the equation holds true, then congratulations! You've successfully solved for 'x'. If not, don't worry! It just means we need to go back and review our steps to see where we might have made a mistake. Solving for 'x' is like finding the treasure at the end of a mathematical journey, and it's a super satisfying feeling when you finally get there.

Example: Solving the Equation

Let's put our newfound skills to the test with a concrete example. We'll take the equation provided and walk through each step together. This will not only solidify our understanding but also show how these steps work in practice. Remember, math is like riding a bike – you need to practice to get better. So, let's dive into the equation and see how we can conquer it. We'll start by clearing the fractions, then simplify, isolate 'x', and finally, solve for 'x'. I'll explain each step in detail, highlighting the key concepts and techniques involved. Think of this as a guided tour through the world of equation solving. By the end of this example, you'll feel much more confident in your ability to tackle similar problems. So, let's grab our mathematical tools and get started on this exciting journey!

Step 5: Checking Your Solution

We've solved for 'x', but our job isn't quite done yet. It's crucial to check your solution to make sure it's correct. This step is like proofreading your work before submitting it – it catches any potential errors and ensures accuracy. To check our solution, we'll substitute the value we found for 'x' back into the original equation. Then, we'll simplify both sides of the equation separately. If the two sides are equal, then our solution is correct! It's like a mathematical handshake – both sides need to agree for the solution to be valid. If the two sides are not equal, then we've made a mistake somewhere, and we need to go back and review our steps. Checking your solution is a vital part of the problem-solving process. It not only verifies your answer but also deepens your understanding of the equation. So, let's make it a habit to always check our solutions – it's the mark of a true math whiz!

Conclusion

And there you have it! We've successfully navigated the world of fraction equations and learned how to find the value of 'x'. We started by understanding the equation, then we cleared the fractions, simplified, isolated 'x', and finally, solved for 'x'. We even learned the importance of checking our solution to ensure accuracy. Solving equations is a fundamental skill in mathematics, and it opens the door to more advanced concepts. So, the next time you encounter a fraction equation, remember the steps we've discussed, and don't be afraid to tackle it head-on. With practice and patience, you'll become a master equation solver in no time! Remember, math is not about memorizing formulas, it's about understanding the process and applying it logically. So, keep practicing, keep exploring, and most importantly, keep having fun with math!