Solving ((4x-9)/3)+2=3(x-2): A Step-by-Step Guide

Hey guys! Let's dive into solving this equation step-by-step. I know math can sometimes seem like a puzzle, but we'll break it down together so it's super clear. This equation, ((4x-9)/3)+2=3(x-2), looks a bit complex at first glance, but trust me, we'll simplify it like pros.

Understanding the Equation

To begin, let’s understand what the equation is asking us to do. We need to find the value of 'x' that makes both sides of the equation equal. Think of it like a balancing act; whatever we do to one side, we have to do to the other to keep things balanced. Our main goal here is to isolate 'x' on one side of the equation. We'll achieve this by performing algebraic operations, which are just fancy ways of saying we'll add, subtract, multiply, and divide in a strategic manner.

When you first see an equation like this, it's like looking at a roadmap. There are multiple paths we can take, but some are more efficient than others. In this case, we have fractions and parentheses, which are like little roadblocks. The best strategy is often to clear these roadblocks early on. This makes the rest of the journey much smoother and reduces the chance of making errors. Believe me, keeping things simple is the name of the game!

Remember, each step we take is a logical deduction. We're not just randomly moving numbers around; we're applying mathematical rules to simplify the equation. So, let's roll up our sleeves and get started. By the end of this article, you'll not only know the answer but also understand the process. That's the real victory – not just getting the right number, but knowing why it's the right number. Understanding the 'why' makes you a math whiz in the long run.

Step 1: Distribute on Both Sides

Okay, so our first mission is to get rid of those pesky parentheses on the right side of the equation. We're going to use something called the distributive property. This basically means we multiply the term outside the parentheses by each term inside. It's like making sure everyone in the group gets their fair share.

On the right side, we have 3(x-2). We need to distribute the 3 across both x and -2. So, 3 times x is 3x, and 3 times -2 is -6. This transforms the right side of the equation from 3(x-2) to 3x - 6. See how we're simplifying things already?

Now, let's talk about the left side, ((4x-9)/3)+2. There's no immediate distribution we can do here, but we have a fraction. Fractions can be a bit of a headache, so we'll tackle that later. For now, let's just rewrite the entire equation with the simplified right side. This gives us: ((4x-9)/3)+2 = 3x - 6.

We’ve made a solid start! By distributing, we've made the equation a bit more manageable. Now, before we move on, take a moment to appreciate what we've done. This step is crucial because it sets the stage for the rest of the solution. Each step we take is a building block, and we're constructing our way to the answer. Feel good about the progress we're making! Math is all about breaking down complex problems into smaller, solvable steps.

Step 2: Eliminate the Fraction

Alright, now let's tackle that fraction on the left side of the equation. Fractions can sometimes make things look more complicated than they really are, so our goal here is to get rid of it. We can do this by multiplying every term in the equation by the denominator of the fraction, which in this case is 3. Think of it like clearing a table – we want to remove anything that's cluttering our workspace.

So, we're going to multiply both sides of the equation ((4x-9)/3)+2 = 3x - 6 by 3. Let's break it down. When we multiply ((4x-9)/3) by 3, the 3 in the numerator and the 3 in the denominator cancel each other out. This leaves us with just 4x - 9. Awesome, right? We've successfully eliminated the fraction from that term.

Next, we multiply 2 by 3, which gives us 6. On the right side, we multiply 3x by 3, resulting in 9x, and we multiply -6 by 3, which gives us -18. So, after multiplying every term by 3, our equation now looks like this: 4x - 9 + 6 = 9x - 18.

See how much cleaner the equation looks now? By eliminating the fraction, we've made it much easier to work with. This step is a game-changer because it simplifies the equation and reduces the chances of making mistakes later on. Remember, the goal is to make the problem as straightforward as possible. We’re not just solving for 'x'; we’re becoming equation-conquering masters!

Step 3: Combine Like Terms

Now that we've gotten rid of the fraction, let's simplify things further by combining like terms. This is like sorting your laundry – we're grouping the similar items together to make things more organized. In our equation 4x - 9 + 6 = 9x - 18, we have constant terms (numbers without 'x') on both sides that we can combine.

On the left side, we have -9 and +6. When we combine these, we get -3 (because -9 + 6 = -3). So, the left side of the equation simplifies to 4x - 3. On the right side, we only have one constant term, -18, so we'll leave that as it is. Our equation now looks like this: 4x - 3 = 9x - 18.

By combining like terms, we've made our equation even simpler. This is a crucial step because it reduces the number of terms we have to deal with, making the rest of the solution process much more manageable. It's like decluttering your room – once you've gotten rid of the unnecessary stuff, you have more space to move around and work effectively.

We’re making excellent progress! Remember, each step we take is bringing us closer to our goal of isolating 'x'. Math is all about simplifying and organizing, and that's exactly what we're doing here. Let's keep this momentum going and move on to the next step!

Step 4: Isolate the Variable Term

Okay, guys, our next mission is to gather all the 'x' terms on one side of the equation and all the constant terms on the other side. This is like separating the ingredients for a recipe – we want to group the like terms together so we can work with them more easily. We have the equation 4x - 3 = 9x - 18, and we want to get all the 'x' terms on one side.

Let's choose to move the 4x term from the left side to the right side. We can do this by subtracting 4x from both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep the equation balanced. So, we subtract 4x from both 4x - 3 and 9x - 18. This gives us: 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've successfully moved the 'x' term to the right side!

Now, let's move the constant term -18 from the right side to the left side. We can do this by adding 18 to both sides of the equation. This gives us: -3 + 18 = 5x - 18 + 18. On the left side, -3 + 18 simplifies to 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 getting so close to solving for 'x'! By isolating the variable term and the constant terms, we've set ourselves up for the final step. Remember, the key to solving equations is to systematically isolate the variable. We're doing a fantastic job of breaking down this problem into manageable steps. Let's keep going!

Step 5: Solve for x

Alright, we've reached the final showdown! We're in the home stretch now, guys. We have the equation 15 = 5x, and our mission is to isolate 'x' completely. This means we need to get 'x' all by itself on one side of the equation.

Currently, 'x' is being multiplied by 5. To undo this multiplication, we need to do the opposite operation, which is division. So, we're going to divide both sides of the equation by 5. This gives us: 15 / 5 = (5x) / 5.

On the left side, 15 / 5 simplifies to 3. On the right side, (5x) / 5 simplifies to just x because the 5 in the numerator and the 5 in the denominator cancel each other out. So, our equation now looks like this: 3 = x.

We did it! We've successfully solved for 'x'. The solution to the equation ((4x-9)/3)+2=3(x-2) is x = 3. Give yourselves a pat on the back – you've tackled a complex equation and come out victorious!

Conclusion

Solving equations like ((4x-9)/3)+2=3(x-2) might seem daunting at first, but as we've seen, it's all about breaking the problem down into manageable steps. We started by distributing to eliminate parentheses, then we cleared the fraction by multiplying every term by the denominator. Next, we combined like terms to simplify the equation further. We isolated the variable term by moving 'x' terms to one side and constant terms to the other. Finally, we solved for 'x' by dividing both sides by the coefficient of 'x'.

Each step was a logical progression, and by following these steps, we were able to find the solution x = 3. Remember, math is like a puzzle – each piece fits together to create the whole picture. The more you practice, the better you'll become at recognizing patterns and solving these puzzles.

So, don't be intimidated by complex equations. Break them down, stay organized, and remember the basic principles. You've got this! Keep practicing, keep learning, and you'll become a math master in no time. And hey, if you ever get stuck, just remember the steps we've covered here. You've now got a solid foundation for solving all sorts of equations. Keep up the great work!