Solve ((4x-9)/3)+2=3(x-2): Step-by-Step Solution

Hey guys! Math can sometimes feel like a puzzle, but don't worry, we're going to break down this equation step-by-step so it feels super easy. We're tackling the equation ((4x-9)/3)+2=3(x-2), and by the end of this guide, you'll not only know how to solve it, but you'll also understand why each step is important. Let's dive in and make math a little less intimidating and a lot more fun!

Understanding the Equation ((4x-9)/3)+2=3(x-2)

Before we jump into the solution, let's first understand what the equation ((4x-9)/3)+2=3(x-2) is all about. This is a linear equation in one variable, 'x'. That simply means we're trying to find the value of 'x' that makes both sides of the equation equal. Think of it like a balanced scale; our goal is to keep the scale balanced throughout the process. The left side of the equation is ((4x-9)/3)+2, which involves a fraction and addition. The right side is 3(x-2), which includes multiplication and subtraction. To solve this, we'll need to use the order of operations (PEMDAS/BODMAS) in reverse to isolate 'x'. This involves getting rid of the parentheses, fractions, and constants step by step until we have 'x' alone on one side. Remember, every step we take must maintain the balance, so whatever we do to one side, we must also do to the other. Understanding this fundamental principle is key to solving any algebraic equation successfully. When you look at equations like this, don't get overwhelmed by the symbols; instead, see it as a journey where each step brings you closer to the solution. We'll simplify each part, using properties of equality, and gradually reveal the value of 'x'. Keep in mind that math equations are like stories, and each symbol and operation tells a part of that story. Our job is to decode that story and find the ending – the value of 'x'. With this understanding, let's get started on our step-by-step journey to solve this equation. We'll break it down into manageable chunks, making sure to explain each step clearly. So, buckle up, and let's make some math magic happen!

Step 1: Distribute on Both Sides

Okay, so the first thing we need to do to solve the equation ((4x-9)/3)+2=3(x-2) is to deal with any parentheses by distributing. On the right side of the equation, we have 3(x-2). To distribute, we multiply 3 by both terms inside the parentheses. So, 3 times x is 3x, and 3 times -2 is -6. This means 3(x-2) becomes 3x - 6. Now, let's rewrite the equation with this simplification: ((4x-9)/3) + 2 = 3x - 6. We've taken our first step in simplifying the equation, and it's already looking a bit cleaner! Distribution is such a crucial step in solving equations. It helps us to unravel the terms and make them more manageable. Imagine if we didn't distribute – we'd be stuck trying to deal with the parentheses, which can make things super complicated. By distributing, we're essentially freeing up the terms so we can work with them individually. Think of it like opening a present; you need to unwrap it to see what's inside! Now, you might be wondering, why did we only distribute on the right side? Well, the left side has a fraction, and we'll tackle that in the next step. For now, we've successfully distributed and simplified one part of the equation. This step is all about applying the distributive property, which states that a(b + c) = ab + ac. It's a fundamental rule in algebra, and mastering it will help you solve a wide range of equations. Remember, math is like building with blocks; each step is a block that adds to the bigger picture. So, let's keep building and move on to the next step. You're doing great so far, guys! Keep up the awesome work!

Step 2: Clear the Fraction

Now that we've distributed, let's tackle that fraction on the left side of the equation ((4x-9)/3)+2=3(x-2). We have ((4x-9)/3) + 2 = 3x - 6. To get rid of the fraction, we need to multiply every term in the equation by the denominator, which is 3 in this case. This will clear the fraction and make our equation much easier to work with. So, we multiply both sides of the equation by 3: 3 * [((4x-9)/3) + 2] = 3 * (3x - 6). When we distribute the 3 on the left side, we get 3 * ((4x-9)/3) + 3 * 2. The 3 in the numerator and the 3 in the denominator of the first term cancel each other out, leaving us with just (4x - 9). And 3 * 2 is 6. So, the left side becomes 4x - 9 + 6. On the right side, we multiply 3 by (3x - 6), which gives us 9x - 18. Now our equation looks like this: 4x - 9 + 6 = 9x - 18. See how much simpler it looks without the fraction? Clearing the fraction is such a powerful technique in algebra. Fractions can often make equations look intimidating, but by multiplying through by the denominator, we eliminate them and create a smoother path to the solution. It's like smoothing out a bumpy road so you can drive more easily. This step is all about using the multiplication property of equality, which states that if you multiply both sides of an equation by the same number, the equation remains balanced. We're maintaining the balance while making the equation more manageable. Think of it like scaling up a recipe; you need to multiply all the ingredients by the same amount to keep the proportions right. Math equations are the same; we need to apply the same operation to both sides to maintain the equality. So, we've successfully cleared the fraction and simplified our equation further. We're making great progress, guys! Let's keep going and see what the next step brings. You're doing an amazing job!

Step 3: Combine Like Terms

Alright, we've cleared the fraction, and our equation ((4x-9)/3)+2=3(x-2) now looks like this: 4x - 9 + 6 = 9x - 18. The next step is to combine like terms on each side of the equation. Like terms are terms that have the same variable raised to the same power (or just constants). On the left side, we have -9 and +6, which are both constants. We can combine them by adding them together: -9 + 6 = -3. So, the left side simplifies to 4x - 3. The right side, 9x - 18, already has its terms simplified, so we don't need to do anything there yet. Now our equation looks even cleaner: 4x - 3 = 9x - 18. Combining like terms is like organizing your closet – you group similar items together to make things neater and easier to find. In math, it simplifies the equation and makes it easier to see the next steps. This is all about the commutative and associative properties of addition, which allow us to rearrange and group terms without changing the value of the expression. These properties are super helpful for simplifying equations. Think of it like rearranging furniture in a room; you can move things around to create a more comfortable space without changing the room itself. Math equations are the same; we can rearrange and group terms to make the equation more manageable. By combining like terms, we're reducing the number of terms in the equation, which makes it easier to isolate the variable 'x'. This is a key step in solving any algebraic equation. The fewer terms we have, the closer we are to finding the solution. So, we've successfully combined like terms and made our equation even simpler. We're really making progress, guys! Let's move on to the next step and continue our journey to solving for 'x'. You're doing fantastic!

Step 4: Isolate the Variable Term

We're cruising along nicely! Our equation ((4x-9)/3)+2=3(x-2) is currently at 4x - 3 = 9x - 18. Now, we need to isolate the variable term, which means getting all the 'x' terms on one side of the equation. To do this, let's subtract 4x from both sides. This will move the 'x' term from the left side to the right side. So, we have 4x - 3 - 4x = 9x - 18 - 4x. On the left side, 4x - 4x cancels out, leaving us with just -3. On the right side, 9x - 4x simplifies to 5x. So, our equation now looks like this: -3 = 5x - 18. We're one step closer to isolating 'x'! Isolating the variable term is like separating the ingredients you need for a recipe from the rest of your pantry. You're gathering all the 'x' terms in one place so you can focus on them. This step utilizes the subtraction property of equality, which states that if you subtract the same quantity from both sides of an equation, the equation remains balanced. We're keeping the balance while moving the 'x' terms where we want them. Think of it like moving puzzle pieces around on a table; you're rearranging them to get closer to the final picture. Math equations are the same; we're rearranging terms to get closer to the solution. By isolating the variable term, we're simplifying the equation and making it easier to solve for 'x'. This is a crucial step in the process. The more we isolate 'x', the closer we get to finding its value. So, we've successfully isolated the variable term and made our equation even more manageable. We're doing a great job, guys! Let's keep the momentum going and move on to the next step. You're rocking this!

Step 5: Isolate the Constant Term

We're getting super close to the finish line! Our equation ((4x-9)/3)+2=3(x-2) has been simplified to -3 = 5x - 18. Now, we need to isolate the constant term on the left side, which means getting rid of the -18 on the right side. To do this, we'll add 18 to both sides of the equation. This will cancel out the -18 on the right side and leave us with just the 'x' term. So, we have -3 + 18 = 5x - 18 + 18. On the left side, -3 + 18 equals 15. On the right side, -18 + 18 cancels out, leaving us with just 5x. So, our equation now looks like this: 15 = 5x. We're almost there! Isolating the constant term is like clearing the clutter around the main object you want to focus on. You're making sure that only the 'x' term is left on one side so you can solve for it. This step uses the addition property of equality, which states that if you add the same quantity to both sides of an equation, the equation remains balanced. We're maintaining the balance while isolating the variable. Think of it like balancing a seesaw; you need to add or remove weight on one side to keep it level. Math equations are the same; we need to apply the same operation to both sides to maintain the equality. By isolating the constant term, we're simplifying the equation even further and making it crystal clear what we need to do next. This is a vital step in solving for 'x'. The closer we get to having 'x' alone on one side, the closer we are to the solution. So, we've successfully isolated the constant term and made our equation incredibly simple. We're on the verge of solving it, guys! Let's move on to the final step and claim our victory. You're doing awesome!

Step 6: Solve for x

We've reached the final step! Our equation ((4x-9)/3)+2=3(x-2) has been simplified to 15 = 5x. To solve for x, we need to get x completely alone on one side of the equation. Since x is being multiplied by 5, we'll do the opposite operation: divide both sides by 5. So, we have 15 / 5 = 5x / 5. On the left side, 15 divided by 5 is 3. On the right side, 5x divided by 5 is just x. So, our equation now looks like this: 3 = x. And there you have it! We've solved for x. The solution is x = 3. Solving for x is like finding the missing piece of a puzzle. You've gone through all the steps, and now you have the final answer. This step utilizes the division property of equality, which states that if you divide both sides of an equation by the same non-zero number, the equation remains balanced. We're maintaining the balance while isolating x. Think of it like sharing a pizza equally among friends; you need to divide the pizza into the right number of slices so everyone gets their fair share. Math equations are the same; we need to divide both sides by the same number to maintain the equality. By solving for x, we've completed the journey and found the value that makes the equation true. This is the ultimate goal of solving any algebraic equation. We've taken a complex-looking equation and broken it down into manageable steps, and now we have the solution. You've done an incredible job, guys! Give yourselves a pat on the back. You've successfully solved for x and conquered this math challenge. Remember, math is a journey, and every step you take brings you closer to understanding the world around you. Keep practicing, keep exploring, and keep the math magic alive!

Conclusion

Woo-hoo! We did it! We successfully solved the equation ((4x-9)/3)+2=3(x-2) by breaking it down into manageable steps. Remember, the key to solving any equation is to take it one step at a time, stay organized, and keep the equation balanced. We started by distributing, then cleared the fraction, combined like terms, isolated the variable term, isolated the constant term, and finally solved for x. And the answer is x = 3! Math might seem daunting at first, but with a clear strategy and a little practice, you can conquer any equation that comes your way. Remember to always double-check your work and understand why each step is necessary. Solving equations is like building a strong foundation for more advanced math concepts. The skills you've learned today will help you tackle even more challenging problems in the future. So, keep practicing, keep exploring, and never be afraid to ask for help. Math is a journey, not a destination, and every step you take makes you a stronger problem-solver. And most importantly, have fun with it! Math can be like a puzzle, and the satisfaction of finding the solution is truly rewarding. You've shown amazing perseverance and dedication throughout this process, and you should be incredibly proud of your accomplishment. Congratulations on solving this equation! You're now one step closer to mastering the world of math. Keep up the fantastic work, guys, and remember that you have the power to solve any mathematical challenge that comes your way. Math is all about logic, patterns, and problem-solving, and these are skills that will benefit you in so many areas of your life. So, keep honing your skills, keep believing in yourself, and keep shining brightly in the world of math. You've got this!