Solving Equations: A Step-by-Step Guide

Hey everyone, let's dive into solving the equation: $\frac{x}{3} + \frac{x-2}{5} = 6$. Don't worry, it might look a bit intimidating at first glance with those fractions, but trust me, we'll break it down step by step and make it super easy to understand. This is a fundamental concept in algebra, and understanding how to solve these types of equations will be incredibly helpful as you move forward in your math journey. We'll be using some basic principles of algebra, like the properties of equality, to isolate the variable 'x' and find its value. So, grab your pencils and let's get started! This guide will provide you with a clear, concise method for tackling this and similar equations. We'll start with the initial problem: $\frac{x}{3} + \frac{x-2}{5} = 6$ and transform it to an easy form that you'll quickly understand. The process of solving equations is a cornerstone of mathematics, and mastering it will open doors to more complex problem-solving. By the end, you'll feel confident in tackling similar equations. So, let’s get started.

Firstly, we must eliminate the fractions to simplify the equation. The presence of fractions can often complicate the solving process. To get rid of these fractions, we need to find the least common multiple (LCM) of the denominators, which are 3 and 5. The LCM of 3 and 5 is 15. We will then multiply every term in the equation by 15. Doing this will clear out the denominators, allowing us to work with whole numbers instead. This is a key step, as it simplifies the equation significantly. The aim here is to get rid of the fraction, and make the expression easier to handle. Now, let's go step by step and break down the equation. This simplifies the equation significantly, paving the way for easier calculations. Multiplying by the LCM is a common and effective technique used to simplify equations involving fractions. Now that we understand the process, let's apply it to our given equation and simplify it. This step is about streamlining our equation to make it more manageable. Understanding and applying the concept of LCM is very crucial in solving this kind of problem. This is a crucial step towards simplifying the equation.

Step-by-Step Solution

Now, let's roll up our sleeves and solve the equation: $\frac{x}{3} + \frac{x-2}{5} = 6$. We'll carefully go through each step to make sure everyone understands the process. This equation requires a methodical approach, and we'll break it down into manageable chunks. The goal is to isolate 'x' on one side of the equation and find its value. Remember, the core principle is to maintain the equality throughout each step. This means whatever operation we perform on one side of the equation, we must also perform on the other side. This ensures that the balance of the equation is maintained. Let’s not get lost, let's break this down into smaller steps. Every single step is important, and each of them brings us closer to the solution. The aim is to convert the given equation into a simpler form. Understanding each step is crucial for mastering similar problems in the future. Now, let's get into the specifics of how to solve this equation step-by-step. Remember, consistency in following these steps will lead you to the solution!

Step 1: Find the Least Common Multiple (LCM). As we said before, the denominators are 3 and 5. The LCM of 3 and 5 is 15. This is the first and perhaps one of the most important steps. It sets the stage for simplifying the equation by eliminating the fractions. Now that we know our starting point, let’s move forward with solving the equation. The LCM is essential because it allows us to convert the fractions into whole numbers, making the equation much easier to solve. The LCM is the smallest number that is a multiple of all the denominators. Finding the LCM is crucial for our simplification. So, identifying the LCM is the first critical step to solve the problem. Remember, always start with finding the LCM for equations with fractions.

Step 2: Multiply Each Term by the LCM. Multiply every term in the equation by 15. This gives us:

$15 * (\frac{x}{3}) + 15 * (\frac{x-2}{5}) = 15 * 6$. This step is a direct application of the LCM we calculated in the previous step. We're effectively getting rid of the fractions in the equation by multiplying each term by 15. This makes the equation easier to solve. Multiplying each term by the LCM is the key to simplifying the equation. It's a fundamental technique used in algebra to handle fractions in equations. The goal here is to transform the equation into a simpler form without fractions. By multiplying each term by 15, we're making the equation easier to solve. This step streamlines our equation.

Step 3: Simplify. Now, let’s simplify each term. This simplifies the previous equation: $5x + 3(x - 2) = 90$. We've essentially removed the fractions. This makes the equation much more manageable. After multiplying each term by the LCM, the next step is to simplify the equation. This is where we start combining terms and getting closer to isolating 'x'. Let's simplify and make the equation more readable and easy to solve. The goal here is to reduce the complexity of the equation. This makes it easier to work with. Now we are closer to finding the solution. This is where the real work begins, simplifying to reach our final answer.

Step 4: Distribute. Distribute the 3 across the terms in the parentheses: $5x + 3x - 6 = 90$. This step is all about expanding the equation to bring like terms together. We’re getting ready to combine the 'x' terms and isolate them on one side of the equation. This helps us to simplify the equation even further. This is a critical step in which we apply the distributive property. It's a standard algebraic method for handling parentheses in an equation. Make sure you don't miss this step, as it's key to simplifying the equation. This distribution sets us up for combining like terms in the following steps. This ensures that the terms are correctly multiplied and that the balance of the equation is maintained.

Step 5: Combine Like Terms. Combine the 'x' terms: $5x + 3x = 8x$. This simplifies the equation to: $8x - 6 = 90$. This is where we bring the terms together. Combining like terms makes the equation simpler and easier to solve. Now, combining the x terms leads us to further steps. The combining of 'x' terms is an essential step towards isolating the variable. Here we combine the 'x' terms. This simplifies the equation and prepares it for the next steps.

Step 6: Isolate the Variable Term. Add 6 to both sides of the equation to get rid of the constant on the left side: $8x - 6 + 6 = 90 + 6$. This simplifies to: $8x = 96$. This is about getting the 'x' term by itself. Isolate the variable term to isolate the variable. We are moving toward isolating x. This step is about getting the variable term alone on one side of the equation. This is a very important step towards isolating 'x'.

Step 7: Solve for x. Divide both sides by 8 to solve for x: $x = \frac{96}{8}$. This gives us: $x = 12$. Now we are on the final step, and we have our solution! Solving for 'x' means finding the value that satisfies the equation. It's the moment of truth where we find the value of x that makes the original equation true. We have finally reached our solution. This final step isolates 'x', giving us its value. Now that we've gone through each step, we've arrived at our final solution. Now we know what x equals, so it's a success!

Conclusion: The Answer

So, after all those steps, the answer is: $x = 12$. Congratulations! You've successfully solved the equation $\frac{x}{3} + \frac{x-2}{5} = 6$. You've now taken on a fundamental algebra concept and successfully solved a linear equation with fractions. By following the steps, you've learned how to isolate the variable and find its value. Awesome work, and keep up the great job! Congratulations on finishing. You’ve earned it, and with a little more practice, you'll be solving these equations in no time. This skill is incredibly useful in various areas of mathematics. Now that you have the knowledge, you can tackle similar problems. Keep practicing, and you'll get better and better at solving these types of equations. You have successfully solved the equation and learned the process. Great job, and congratulations! You’ve done it!