Solve (x-4)/2 = 3/5: Step-by-Step Guide

Introduction

Hey guys! Today, we're diving into a common type of algebraic equation that you'll often encounter in mathematics: solving equations involving fractions. Specifically, we're going to break down the step-by-step process of solving the equation (x-4)/2 = 3/5. This type of problem is fundamental in algebra, and mastering it will set you up for success in more advanced topics. We'll explore the underlying principles, the practical steps, and some common pitfalls to avoid. Whether you're a student tackling homework, a math enthusiast looking to refresh your skills, or just curious about how these things work, this guide is for you. So, let's jump right in and make this equation crystal clear!

Understanding how to solve algebraic equations is crucial because it forms the backbone of many mathematical and scientific disciplines. Equations like the one we're tackling, (x-4)/2 = 3/5, might seem daunting at first, but they're actually quite straightforward once you understand the basic principles. Think of an equation as a balanced scale: whatever you do to one side, you must also do to the other to maintain the balance. This is the golden rule of equation solving, and we'll use it extensively throughout this guide. The main goal when solving for 'x' is to isolate it on one side of the equation. This means getting 'x' by itself, with no other numbers or operations attached to it. We achieve this by performing inverse operations – operations that undo each other. For example, if 'x' is being divided by 2, we can undo that division by multiplying by 2. If '4' is being subtracted from 'x,' we can undo that subtraction by adding 4. By strategically applying these inverse operations, we can gradually peel away the layers surrounding 'x' until we have it all by itself. This process is not only a fundamental skill in algebra but also a powerful problem-solving technique that can be applied in many other areas of life. So, let’s get started and see how this works in practice!

Step 1: Eliminating the Fractions

Alright, the first thing we need to tackle when solving the equation (x-4)/2 = 3/5 is those pesky fractions! Fractions can sometimes make equations look more complicated than they actually are, so our initial goal is to get rid of them. The most effective way to do this is by using a technique called cross-multiplication. Cross-multiplication is a neat trick that works when you have an equation with a fraction on each side, like ours. It involves multiplying the numerator (the top number) of one fraction by the denominator (the bottom number) of the other fraction. In our case, we multiply (x-4) by 5 and 3 by 2. This is based on the fundamental principle that if two fractions are equal, then their cross-products are also equal. Mathematically, this looks like: a/b = c/d is equivalent to ad = bc. Applying this to our equation, we get 5*(x-4) = 3*2. This step is crucial because it transforms the equation from one involving fractions to a much simpler linear equation, which is easier to solve. By eliminating the fractions, we clear the way for the next steps in isolating 'x'. It's like clearing the underbrush in a forest to get a better view of the trees – we're making the equation more manageable and easier to work with. Remember, the key to mastering algebra is to break down complex problems into simpler, more digestible steps. Eliminating fractions is a perfect example of this strategy. So, with the fractions gone, we're now in a much better position to find the value of 'x'. Let's move on to the next step and see how we can further simplify the equation.

Step 2: Distributing and Simplifying

Now that we've eliminated the fractions in our equation, we're left with 5*(x-4) = 32. The next step is to simplify this equation further, and that's where the distributive property comes into play. The distributive property is a fundamental concept in algebra that allows us to multiply a number by a sum or difference inside parentheses. In simpler terms, it means we need to multiply the number outside the parentheses by each term inside the parentheses. In our case, we need to multiply 5 by both 'x' and '-4'. So, 5(x-4) becomes 5x - 54, which simplifies to 5x - 20. On the other side of the equation, we have 3*2, which is simply 6. So, our equation now looks like this: 5x - 20 = 6. This step is crucial because it expands the expression and allows us to combine like terms, which will eventually help us isolate 'x'. Distributing correctly is like carefully unpacking a package – you need to make sure you handle each item inside properly. A common mistake is forgetting to distribute the 5 to both terms inside the parentheses, so always double-check your work! Once we've correctly distributed and simplified, the equation becomes much cleaner and easier to manage. We've transformed a more complex expression into a simpler linear equation that's just begging to be solved. Now, let's move on to the next step and see how we can further isolate 'x' and get closer to our final answer.

Step 3: Isolating the Variable

Okay, guys, we're making great progress! Our equation is now 5x - 20 = 6. The next key step is to isolate the variable, which means getting 'x' all by itself on one side of the equation. To do this, we need to undo any operations that are being performed on 'x'. In this case, we have two operations to deal with: multiplication by 5 and subtraction of 20. Remember our golden rule of equation solving: whatever we do to one side, we must also do to the other. So, let's start by undoing the subtraction of 20. To do this, we add 20 to both sides of the equation. This gives us: 5x - 20 + 20 = 6 + 20. Simplifying this, we get 5x = 26. Notice how adding 20 to both sides effectively cancels out the -20 on the left side, bringing us closer to isolating 'x'. Now, we have just one operation left to undo: the multiplication by 5. To do this, we divide both sides of the equation by 5. This gives us: (5x)/5 = 26/5. Simplifying this, we get x = 26/5. And there you have it! We've successfully isolated 'x' and found its value. Isolating the variable is like carefully extracting a precious gem from a mine – you need to be precise and methodical in your approach. Each step we take brings us closer to the final solution. So, we've found that x = 26/5, but before we declare victory, let's move on to the final step: checking our answer.

Step 4: Checking the Solution

Alright, we've arrived at a potential solution: x = 26/5. But before we celebrate, it's crucial to check our solution. This step is like proofreading a document before submitting it – it ensures that we haven't made any mistakes along the way. Checking our solution involves substituting the value we found for 'x' back into the original equation and seeing if it holds true. Our original equation was (x-4)/2 = 3/5. So, we're going to replace 'x' with 26/5 and see if both sides of the equation are equal. Substituting, we get ((26/5) - 4)/2 = 3/5. Now, let's simplify the left side of the equation. First, we need to subtract 4 from 26/5. To do this, we need to express 4 as a fraction with a denominator of 5, which is 20/5. So, we have (26/5) - (20/5) = 6/5. Now, our equation looks like this: (6/5)/2 = 3/5. Next, we need to divide 6/5 by 2. Dividing by 2 is the same as multiplying by 1/2, so we have (6/5) * (1/2) = 6/10. We can simplify 6/10 by dividing both the numerator and denominator by 2, which gives us 3/5. So, the left side of the equation simplifies to 3/5, which is exactly the same as the right side of the equation! This confirms that our solution, x = 26/5, is correct. Checking your solution is a vital step in any math problem, and it's especially important in algebra. It gives you confidence that your answer is accurate and helps you catch any errors you might have made. It’s like having a safety net – it prevents you from falling into the trap of an incorrect answer. So, always make sure to check your solutions, guys! With our solution checked and verified, we can confidently say that we've successfully solved the equation.

Conclusion

Awesome job, guys! We've walked through the entire process of solving the equation (x-4)/2 = 3/5, from eliminating fractions to checking our solution. We've seen how cross-multiplication helps us get rid of fractions, how the distributive property allows us to simplify expressions, and how isolating the variable leads us to the answer. We also emphasized the importance of checking our solution to ensure accuracy. This type of equation is a building block for more advanced algebraic concepts, so mastering it is super important. Remember, practice makes perfect! The more you work through these types of problems, the more comfortable and confident you'll become. Don't be afraid to make mistakes – they're part of the learning process. Just make sure to learn from them and keep pushing forward. Whether you're tackling homework, preparing for a test, or just curious about math, the skills we've covered today will definitely come in handy. So, keep practicing, keep exploring, and keep having fun with math! If you ever get stuck, remember the steps we've outlined, and don't hesitate to review this guide or seek help from a teacher or tutor. You've got this!