Solve For H: Step-by-Step Guide To 17 - 4h = 4h - 15

Hey guys! Let's dive into a fun math problem today. We're going to tackle the equation 17 - 4h = 4h - 15 and figure out how to solve for h. Don't worry, it's not as scary as it looks! We'll break it down step by step so it's super easy to follow. Whether you're a math whiz or just trying to brush up on your algebra skills, this guide will help you understand the process. So, grab your pencil and paper, and let's get started!

Understanding the Basics of Algebraic Equations

Before we jump into solving for h in the equation 17 - 4h = 4h - 15, it's super important to get a good grasp of the basics of algebraic equations. Think of an algebraic equation like a balanced scale. On one side, you have an expression, and on the other side, you have another expression. The goal is to keep the scale balanced while you're working to isolate the variable – in this case, h. This means whatever operation you perform on one side of the equation, you must perform the exact same operation on the other side. This principle ensures that the equality remains true throughout the solving process. We are essentially manipulating the equation while maintaining its inherent balance.

Let's talk about the different parts of our equation. We have constants, which are the numbers that stand alone, like 17 and -15 in our equation. Then we have variables, which are the letters that represent unknown values – in our case, h. And we have coefficients, which are the numbers multiplied by the variables, such as -4 and 4 in our equation. Understanding these components is crucial because it allows us to strategically move terms around and simplify the equation. For instance, we often aim to group like terms together, meaning constants with constants and variables with variables. This makes the equation cleaner and easier to solve.

When we say we're solving for h, what we really mean is that we want to isolate h on one side of the equation. In other words, we want to get h all by itself, with a coefficient of 1 (which we usually don't write explicitly). To do this, we use inverse operations. Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations, and multiplication and division are inverse operations. By applying these inverse operations strategically, we can peel away the layers surrounding h until it stands alone, revealing its value. So, with these foundational concepts in mind, we’re well-prepared to tackle the equation 17 - 4h = 4h - 15 and find the solution for h!

Step 1: Grouping Like Terms – Moving the 'h' Terms

The first key step in solving the equation 17 - 4h = 4h - 15 is to group the like terms together. What does this mean, guys? It means we want to get all the terms that contain our variable, h, on one side of the equation, and all the constant terms (the numbers without any variables) on the other side. This makes the equation much easier to work with and brings us closer to isolating h.

In our equation, we have -4h on the left side and 4h on the right side. To get all the h terms together, we can choose to move either the -4h or the 4h. A common strategy is to move the term that will result in a positive coefficient for h, as this simplifies the subsequent steps. In this case, it makes sense to move the -4h from the left side to the right side. How do we do that? Remember the golden rule of equations: whatever you do to one side, you must do to the other to maintain the balance.

Since we have -4h on the left, we perform the inverse operation, which is adding 4h. So, we add 4h to both sides of the equation:

17 - 4h + 4h = 4h - 15 + 4h

On the left side, -4h + 4h cancels out, leaving us with just 17. On the right side, 4h + 4h combines to give us 8h. So, our equation now looks like this:

17 = 8h - 15

See how much simpler that looks already? We've successfully grouped the h terms on one side. Now, we need to deal with the constant terms to further isolate h. Grouping like terms is a fundamental technique in algebra, and mastering it will make solving equations like this one much smoother. Next up, we'll tackle those constant terms and keep moving towards our solution!

Step 2: Grouping Like Terms – Moving the Constant Terms

Alright, we've made great progress by grouping the h terms together in our equation 17 = 8h - 15. Now, it's time to focus on the constant terms. Remember, our goal is to isolate h on one side of the equation, so we need to get all the numbers without h on the other side. Currently, we have 17 on the left side and -15 on the right side. We need to move that -15 to the left side to join the 17. How do we do that?

Just like before, we use the concept of inverse operations. Since we have -15 on the right side, we perform the inverse operation, which is adding 15. And remember, whatever we do to one side of the equation, we must do to the other side to keep the equation balanced. So, we add 15 to both sides:

17 + 15 = 8h - 15 + 15

On the left side, 17 + 15 equals 32. On the right side, -15 + 15 cancels out, leaving us with just 8h. So, our equation now looks like this:

32 = 8h

We're getting closer! We've successfully grouped all the constant terms on the left side and we have our h term all by itself on the right. This is a big step because the equation is now in a much simpler form. We're almost there, guys! We just have one more step to perform to completely isolate h and find its value. Keep up the great work!

Step 3: Isolating 'h' – Division

We've arrived at the final step in solving for h in the equation 32 = 8h. We've grouped the like terms, and now we have a simplified equation. What's left to do? We need to isolate h completely. Currently, h is being multiplied by 8. To get h by itself, we need to undo this multiplication. Can you guess what operation we'll use?

That's right, we'll use the inverse operation of multiplication, which is division. We need to divide both sides of the equation by the coefficient of h, which is 8. Remember the golden rule: whatever you do to one side of the equation, you must do to the other side to maintain the balance. So, we divide both sides by 8:

32 / 8 = 8h / 8

On the left side, 32 / 8 equals 4. On the right side, 8h / 8 simplifies to just h, because 8 divided by 8 is 1, and 1 times h is simply h. So, our equation now looks like this:

4 = h

Or, we can write it as:

h = 4

We did it! We've successfully isolated h and found its value. The solution to the equation 17 - 4h = 4h - 15 is h = 4. Pat yourselves on the back, guys! You've navigated through the steps of solving an algebraic equation, and you've reached the answer. Now, let's take a moment to check our work and make sure we've got it right.

Step 4: Checking the Solution

Alright, we've solved for h and found that h = 4. But before we confidently say we've nailed it, it's super important to check our solution. Why? Because checking our work helps us catch any potential mistakes we might have made along the way. It's like a safety net that ensures our answer is correct. So, how do we check our solution?

We plug the value we found for h (which is 4) back into the original equation: 17 - 4h = 4h - 15. We'll substitute 4 wherever we see h and then simplify both sides of the equation. If both sides end up being equal, then our solution is correct. If they don't, it means we need to go back and review our steps to find the error.

Let's plug in h = 4 into the original equation:

17 - 4(4) = 4(4) - 15

Now, we simplify each side. On the left side, we have:

17 - 4(4) = 17 - 16 = 1

On the right side, we have:

4(4) - 15 = 16 - 15 = 1

So, we have:

1 = 1

Look at that! Both sides of the equation are equal. This confirms that our solution, h = 4, is indeed correct. We've successfully solved the equation and verified our answer. Checking your solution is a crucial step in problem-solving, guys. It gives you confidence in your answer and helps you avoid making careless mistakes. So, always remember to check your work whenever you can!

Conclusion: Mastering Algebraic Equations

Awesome job, guys! We've successfully solved the equation 17 - 4h = 4h - 15 and found that h = 4. We also took the extra step to check our solution, which confirmed that we got it right. By breaking down the problem into manageable steps – grouping like terms and using inverse operations – we were able to isolate the variable h and find its value. This process is the foundation of solving many algebraic equations, and you've now got a solid understanding of how it works.

Solving algebraic equations might seem challenging at first, but with practice, it becomes much easier. The key is to understand the underlying principles, like maintaining the balance of the equation and using inverse operations. Remember, algebra is like a puzzle, and each equation is a new challenge to solve. The more you practice, the better you'll become at recognizing patterns and applying the right techniques.

So, what's the takeaway here? You've learned how to solve for a variable in a linear equation, and you've seen the importance of checking your work. These are valuable skills that will help you in many areas of math and beyond. Keep practicing, keep challenging yourselves, and don't be afraid to ask for help when you need it. You've got this, guys! Keep up the amazing work, and you'll be mastering algebraic equations in no time!