Solving (2x)/9 - 1/6 = (3x)/18 + 1: A Step-by-Step Guide

Hey guys! Let's break down this math problem together. We've got the equation (2x)/9 - 1/6 = (3x)/18 + 1, and our mission is to find the value of 'x' that makes this equation true. Don't worry, it might look a bit intimidating at first, but we'll take it step by step and you'll see it's totally manageable. We're going to dive deep into the world of algebraic equations, tackling fractions and variables like pros. So, grab your calculators and let's get started!

Understanding the Equation

Before we jump into solving, let's make sure we fully understand what we're dealing with. Our equation is (2x)/9 - 1/6 = (3x)/18 + 1. This is a linear equation, which means it involves a variable (in this case, 'x') raised to the power of 1. The goal is to isolate 'x' on one side of the equation to find its value. We have fractions involved, which might seem tricky, but don't fret! We'll use a common denominator to simplify things. Understanding the structure of the equation is key to solving it efficiently. We need to identify the different terms, the variable, and the constants. This initial analysis sets the stage for the rest of our solution. Remember, math isn't just about getting the right answer; it's about understanding the process. By breaking down the equation into its components, we can approach it with confidence and clarity. Think of it like building a house – you need a solid foundation before you can start adding the walls and roof. In this case, our foundation is a clear understanding of the equation's structure. So, let's take a deep breath and appreciate the beauty of this algebraic puzzle before we start piecing it together.

Finding a Common Denominator

The first hurdle we need to overcome is those pesky fractions. To combine them, we need a common denominator. Looking at our denominators (9, 6, and 18), the least common multiple is 18. This means we'll rewrite each fraction with a denominator of 18. So, let's get to it! We need to convert (2x)/9 and 1/6 into equivalent fractions with a denominator of 18. To convert (2x)/9, we multiply both the numerator and denominator by 2, giving us (4x)/18. For 1/6, we multiply both the numerator and denominator by 3, resulting in 3/18. Now our equation looks like this: (4x)/18 - 3/18 = (3x)/18 + 1. See? It's already looking cleaner! Finding a common denominator is a fundamental step in solving equations with fractions. It allows us to combine like terms and simplify the equation. Think of it like comparing apples and oranges – you need to convert them to the same unit (like pieces of fruit) before you can add them together. In this case, our common unit is a denominator of 18. By finding this common ground, we can proceed with the next steps of solving the equation with much more ease and confidence. This step is not just about the mechanics of math; it's about creating a level playing field where we can compare and combine terms effectively.

Simplifying the Equation

Now that we have a common denominator, let's simplify things further. We can combine the fractions on the left side of the equation: (4x)/18 - 3/18 becomes (4x - 3)/18. Our equation now looks like this: (4x - 3)/18 = (3x)/18 + 1. Much better, right? Next, let's get rid of the fractions altogether! We can do this by multiplying both sides of the equation by 18. This cancels out the denominators, leaving us with 4x - 3 = 3x + 18. Awesome! We've transformed our equation into a much simpler form, without any fractions in sight. Simplifying the equation is a crucial step in the problem-solving process. It's like decluttering your workspace before starting a project – it removes distractions and allows you to focus on the essential elements. By combining like terms and eliminating fractions, we've made our equation more manageable and easier to solve. This step also highlights the power of algebraic manipulation. By applying the same operation to both sides of the equation, we maintain its balance while transforming it into a more convenient form. So, let's appreciate the elegance of this simplification process as we move closer to finding the value of 'x'.

Isolating the Variable 'x'

Our next goal is to isolate 'x' on one side of the equation. We have 4x - 3 = 3x + 18. Let's subtract 3x from both sides to get the 'x' terms together: 4x - 3x - 3 = 3x - 3x + 18, which simplifies to x - 3 = 18. Now, to get 'x' completely alone, we'll add 3 to both sides: x - 3 + 3 = 18 + 3, resulting in x = 21. Boom! We've found our solution! Isolating the variable is the heart of solving algebraic equations. It's like performing surgery to extract the unknown from a complex system. By strategically adding or subtracting terms from both sides of the equation, we gradually strip away the layers surrounding 'x' until it stands alone, revealing its true value. This process requires careful attention to detail and a clear understanding of algebraic principles. Each step brings us closer to the solution, like peeling away the petals of a flower to reveal its center. So, let's celebrate this moment of triumph as we successfully isolate 'x' and uncover its hidden value. This is where the puzzle pieces finally click into place, and the equation surrenders its secret.

Checking the Solution

It's always a good idea to check our answer to make sure it's correct. We found that x = 21. Let's plug this value back into our original equation: (2 * 21)/9 - 1/6 = (3 * 21)/18 + 1. Simplifying the left side, we get 42/9 - 1/6. To subtract these fractions, we need a common denominator, which is 18. So, we have (42 * 2)/18 - (1 * 3)/18 = 84/18 - 3/18 = 81/18. Now, let's simplify the right side: (3 * 21)/18 + 1 = 63/18 + 1. To add 1, we need to express it as a fraction with a denominator of 18, so we have 63/18 + 18/18 = 81/18. Both sides of the equation are equal (81/18 = 81/18), so our solution is correct! Checking the solution is like proofreading your work before submitting it – it catches any potential errors and ensures accuracy. By substituting the value of 'x' back into the original equation, we can verify that it satisfies the equation and makes both sides equal. This step provides peace of mind and confirms that our solution is not just a guess, but a valid answer. It also reinforces our understanding of the equation and the solution process. Think of it like testing a bridge before opening it to traffic – you want to make sure it can handle the load. In this case, our 'load' is the value of 'x', and our 'bridge' is the equation. By checking the solution, we're ensuring that our bridge is strong and our answer is sound.

Final Answer

So, after all that work, we've arrived at our final answer: x = 21. We successfully solved the equation (2x)/9 - 1/6 = (3x)/18 + 1. Give yourselves a pat on the back, guys! You've tackled fractions, simplified equations, and isolated variables like true math champions. The final answer is the culmination of our efforts, the reward for our perseverance and problem-solving skills. It's like reaching the summit of a mountain after a challenging climb – the view is breathtaking and the sense of accomplishment is immense. But the journey is just as important as the destination. By working through each step of the solution, we've deepened our understanding of algebraic equations and honed our mathematical abilities. So, let's celebrate our success not just in finding the answer, but in mastering the process. And remember, the skills we've learned today will serve us well in future mathematical adventures.

Solve the equation: (2x)/9 - 1/6 = (3x)/18 + 1

Solving (2x)/9 - 1/6 = (3x)/18 + 1 A Step-by-Step Guide