Solving For H: A Step-by-Step Guide

Hey everyone! Let's dive into solving this equation for 'h': $\frac{2}{3}(9h-3) = 3h$. If you've ever felt a little lost when dealing with algebraic equations, don't worry, we're going to break it down together step by step. We'll go through each stage slowly so you can fully understand it, from removing parentheses to isolating 'h' on one side of the equation. Solving equations like this is a fundamental skill in mathematics, and once you've mastered it, you'll find it super useful in all sorts of other problems. So, let's get started and make math a little less mysterious and a lot more fun!

Understanding the Equation

Before we jump into solving for h, let's take a good look at the equation: $\frac{2}{3}(9h-3) = 3h$. Equations like this one are the bread and butter of algebra, and they might look intimidating at first, but trust me, they're totally manageable once you break them down. The main goal here is to isolate h on one side of the equation. This means we want to get h all by itself so we can see exactly what value makes the equation true. Equations are like puzzles – we have to follow certain rules to solve them, but each step gets us closer to the solution. We need to recall the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), but also keep in mind that when solving, we often reverse this order. Think of it like peeling back the layers of an onion – we need to tackle the outermost layers first to get to the core. In our case, we'll start by dealing with the parentheses and then move on to isolating h. Remember, algebra is all about balance, so whatever we do to one side of the equation, we have to do to the other. This keeps the equation true and helps us maintain that equilibrium as we work towards our solution. So, let's roll up our sleeves and get started – we're about to turn this equation from a puzzle into a piece of cake!

Step 1: Distribute the $\frac{2}{3}$

The first move we're going to make in solving for h is to deal with the parentheses in the equation $\frac{2}{3}(9h-3) = 3h$. To do this, we'll use the distributive property. This property is like a friendly tool in our math toolbox, allowing us to multiply a single term by each term inside the parentheses. In our case, we need to multiply $\frac{2}{3}$ by both $9h$ and $-3$. So, let's break it down. First, we multiply $\frac{2}{3}$ by $9h$. Think of it as $\frac{2}{3} * 9h$. To make it easier, we can write 9h as $\frac{9h}{1}$, so we have $\frac{2}{3} * \frac{9h}{1}$. Multiplying fractions is straightforward – you multiply the numerators (the top numbers) and the denominators (the bottom numbers). So, $2 * 9h = 18h$ and $3 * 1 = 3$. This gives us $\frac{18h}{3}$, which simplifies to $6h$. Next, we multiply $\frac{2}{3}$ by $-3$. Again, we can write $-3$ as $\frac{-3}{1}$, so we have $\frac{2}{3} * \frac{-3}{1}$. Multiplying the numerators gives us $2 * -3 = -6$, and multiplying the denominators gives us $3 * 1 = 3$. This results in $\frac{-6}{3}$, which simplifies to $-2$. Now, let's put it all together. By distributing the $\frac{2}{3}$, we've transformed the left side of the equation from $\frac{2}{3}(9h-3)$ to $6h - 2$. This makes our equation much simpler to work with. Remember, this distribution step is crucial because it allows us to get rid of the parentheses and move closer to isolating h. So, after this step, our equation looks like this: $6h - 2 = 3h$. We're making great progress!

Step 2: Combine Like Terms

Now that we've distributed the $\frac{2}{3}$ and our equation looks like $6h - 2 = 3h$, it's time to gather all the h terms on one side and the constants (the numbers without variables) on the other. Think of it as sorting your socks – you want all the h-socks in one drawer and all the number-socks in another! In this case, we have h terms on both sides of the equation: $6h$ on the left and $3h$ on the right. To combine these, we need to move one of them. It's usually easier to move the smaller one to avoid dealing with negative numbers, but either way will work. Let's subtract $3h$ 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 $3h$ from the left side ($6h - 2 - 3h$) and also from the right side ($3h - 3h$). On the left, $6h - 3h$ simplifies to $3h$, so we now have $3h - 2$. On the right, $3h - 3h$ equals zero, which is exactly what we wanted! Our equation now looks like $3h - 2 = 0$. See how much simpler it's becoming? Now, we need to get the constant term, $-2$, to the other side of the equation. To do this, we'll add 2 to both sides. So, we have $3h - 2 + 2 = 0 + 2$. On the left, $-2 + 2$ cancels out, leaving us with just $3h$. On the right, $0 + 2$ equals 2. Our equation is now $3h = 2$. We're almost there! By combining like terms, we've managed to isolate the h term on one side and the constant on the other. This is a crucial step in solving any algebraic equation, and it sets us up perfectly for the final step.

Step 3: Isolate h

Alright, we've reached the final stretch! Our equation is currently sitting pretty as $3h = 2$. The ultimate goal, as you remember, is to get h all by itself on one side of the equation. This is what we mean by isolating the variable. Right now, h is being multiplied by 3. To undo this multiplication and free h, we need to perform the opposite operation, which is division. So, we're going to divide both sides of the equation by 3. This is a super important step – remember, whatever you do to one side of the equation, you absolutely have to do to the other to keep things balanced and the equation true. Think of it like a scale; if you add or take away weight from one side, you need to do the same on the other to keep it level. So, let's divide. On the left side, we have $3h$ divided by 3, which we can write as $\frac{3h}{3}$. The 3 in the numerator (the top) and the 3 in the denominator (the bottom) cancel each other out, leaving us with just h. This is exactly what we wanted! On the right side, we have 2 divided by 3, which we write as $\frac{2}{3}$. This fraction doesn't simplify any further, so we'll leave it as it is. Putting it all together, we now have $h = \frac{2}{3}$. Boom! We've done it. We've successfully isolated h and found its value. This is the solution to our equation. Give yourself a pat on the back – you've navigated the algebraic terrain like a pro!

Solution

So, after all our hard work, we've arrived at the solution: $h = \frac{2}{3}$. This means that if we substitute $\frac{2}{3}$ for h in the original equation, $\frac{2}{3}(9h-3) = 3h$, both sides of the equation will be equal. It's like finding the missing piece of a puzzle that makes everything fit perfectly. To be absolutely sure we've got it right, it's always a good idea to check our answer. This is like proofreading your work – it helps catch any sneaky mistakes that might have slipped through. To check, we'll plug $\frac{2}{3}$ back into the original equation in place of h. So, we have $\frac{2}{3}(9(\frac{2}{3})-3) = 3(\frac{2}{3})$. Now, we need to simplify each side to see if they're equal. First, let's look at the left side. We have $9 * \frac{2}{3}$, which equals $\frac{18}{3}$, which simplifies to 6. So, we now have $\frac{2}{3}(6-3)$. Inside the parentheses, $6 - 3$ equals 3, so we're left with $\frac{2}{3} * 3$. This equals $\frac{6}{3}$, which simplifies to 2. So, the left side of the equation equals 2. Now, let's tackle the right side: $3(\frac{2}{3})$. This equals $\frac{6}{3}$, which also simplifies to 2. Ta-da! Both sides of the equation equal 2. This confirms that our solution, $h = \frac{2}{3}$, is indeed correct. You've not only solved the equation, but you've also verified your answer – that's fantastic! This step of checking your solution is a habit that will serve you well in all your mathematical adventures. It gives you confidence that you've got the right answer and helps you understand the problem even better.

Conclusion

Great job, everyone! We've successfully navigated the equation $\frac{2}{3}(9h-3) = 3h$ and found that $h = \frac{2}{3}$. You've shown some serious math muscles by working through this problem step by step, from distributing the fraction to isolating h. Remember, solving equations is like learning any new skill – it takes practice, patience, and a bit of persistence. But with each equation you solve, you're building a stronger foundation for more advanced math concepts. The key takeaways from this exercise are the importance of the distributive property, combining like terms, and performing inverse operations to isolate the variable. These are the bread and butter of algebra, and they'll come up again and again in your mathematical journey. Don't be afraid to tackle new problems and apply what you've learned here. And remember, if you ever get stuck, it's okay to ask for help or review the steps. Math is a collaborative adventure, and we're all in this together. So, keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!