Solving For H An Exact Answer Explained
Hey guys! Today, we're going to dive into a super common algebra problem: solving for a specific variable. In this case, we're tackling the equation 1. 5(2 - 4h) = 6h and our mission is to isolate 'h' and find its exact value. No approximations here – we're going for the precise answer! So, grab your pencils, and let's get started on this mathematical adventure together.
Understanding the Equation
Before we jump into solving for 'h', let's break down the equation we're working with: 1. 5(2 - 4h) = 6h. At its core, this is a linear equation, meaning the highest power of our variable 'h' is 1. This makes it relatively straightforward to solve. The equation consists of two sides separated by an equals sign (=). Our goal is to manipulate the equation using algebraic operations until we have 'h' isolated on one side and its value on the other.
The left side of the equation, 1. 5(2 - 4h), involves a distributive property. This means we need to multiply 1.5 by both terms inside the parentheses: 2 and -4h. This step is crucial because it helps us eliminate the parentheses and simplify the equation. Remember, the order of operations (PEMDAS/BODMAS) is our guiding principle here. We handle the multiplication before any addition or subtraction.
The right side of the equation, 6h, is a simple term representing 6 times the variable 'h'. There's nothing to simplify here just yet, but it's important to keep track of this term as we move things around in the equation. The key to solving any equation is maintaining balance. Whatever operation we perform on one side, we must perform the same operation on the other side to keep the equation true. This is the fundamental principle that allows us to isolate 'h'. Think of it like a mathematical seesaw – we need to keep both sides level.
Step-by-Step Solution
Okay, let's get down to the nitty-gritty and solve this equation step by step. Remember, our equation is 1. 5(2 - 4h) = 6h. The first thing we need to do, as we discussed earlier, is apply the distributive property on the left side. This means multiplying 1.5 by both 2 and -4h. So, 1. 5 multiplied by 2 equals 3, and 1.5 multiplied by -4h equals -6h. This transforms our equation into:
3 - 6h = 6h
Now, we have a much simpler-looking equation! The next step is to get all the 'h' terms on one side of the equation. To do this, we can add 6h to both sides. This will eliminate the -6h term on the left side. Adding 6h to both sides gives us:
3 - 6h + 6h = 6h + 6h
Simplifying this, we get:
3 = 12h
We're getting closer! Now, we have 3 equals 12 times 'h'. To isolate 'h', we need to get rid of the 12. Since 'h' is being multiplied by 12, we can do the opposite operation: divide both sides by 12. This gives us:
3 / 12 = 12h / 12
Simplifying this, we get:
h = 3 / 12
We're almost there! The final step is to simplify the fraction 3/12. Both 3 and 12 are divisible by 3. Dividing both the numerator and the denominator by 3, we get:
h = 1 / 4
And there you have it! We've solved for 'h', and the exact answer is 1/4. Woohoo! Isn't it satisfying when everything clicks into place?
Common Mistakes to Avoid
When solving equations like this, there are a few common pitfalls that students often stumble into. Let's highlight these so you can steer clear of them and ace your algebra problems!
One of the most frequent errors is forgetting to distribute properly. Remember, when you have a number multiplied by a quantity in parentheses, you need to multiply that number by every term inside the parentheses. In our example, that means multiplying 1.5 by both 2 and -4h. If you only multiply by one term, you'll end up with the wrong equation and a wrong answer.
Another common mistake is not maintaining balance in the equation. Whatever operation you perform on one side, you must perform the exact same operation on the other side. If you add something to one side but not the other, you've broken the equation, and your solution will be incorrect. Think of it like a seesaw – if you add weight to one side, you need to add the same weight to the other to keep it balanced.
Sign errors are also a sneaky culprit in algebra problems. Be extra careful when dealing with negative signs. Make sure you're applying the correct rules for multiplying and adding negative numbers. For example, a negative times a negative is a positive, and a negative times a positive is a negative. Keeping track of those signs can make a huge difference in your final answer.
Finally, don't forget to simplify your answer as much as possible. In our case, we ended up with the fraction 3/12, which could be simplified to 1/4. Always reduce fractions to their simplest form and combine like terms whenever possible. This not only gives you the most accurate answer but also demonstrates a strong understanding of mathematical principles.
By being mindful of these common mistakes, you can boost your confidence and accuracy when solving algebraic equations. Remember, practice makes perfect, so keep working at it, and you'll become a pro in no time!
Verification
Alright, guys, we've solved for 'h' and gotten an answer of 1/4. But how can we be absolutely sure that our answer is correct? This is where verification comes in handy. Verification is the process of plugging our solution back into the original equation to see if it holds true. It's like a mathematical safety check that gives us peace of mind.
So, let's take our original equation, 1. 5(2 - 4h) = 6h, and substitute 1/4 for 'h'. This gives us:
- 5(2 - 4 * (1/4)) = 6 * (1/4)
Now, we need to simplify both sides of the equation following the order of operations (PEMDAS/BODMAS). Let's start with the left side. Inside the parentheses, we have 4 multiplied by 1/4, which equals 1. So, the expression inside the parentheses becomes 2 - 1, which equals 1. Now the left side looks like this:
- 5 * 1
Which simply equals 1.5.
Now, let's move to the right side of the equation. We have 6 multiplied by 1/4. This equals 6/4, which can be simplified to 3/2 or 1.5.
So, our equation now looks like this:
- 5 = 1.5
And there you have it! Both sides of the equation are equal, which means our solution, h = 1/4, is correct! High five!
Verification is a crucial step in problem-solving. It not only confirms your answer but also helps you catch any errors you might have made along the way. It's a great habit to get into, especially when dealing with more complex equations. So, always take that extra minute to verify your solution – it's worth it!
Conclusion
Fantastic job, everyone! We've successfully solved for 'h' in the equation 1. 5(2 - 4h) = 6h. We walked through the process step-by-step, from understanding the equation to applying the distributive property, isolating 'h', and finally, verifying our solution. We even covered some common mistakes to avoid so you can tackle similar problems with confidence.
The key takeaways from this exercise are the importance of the distributive property, maintaining balance in the equation, being careful with signs, and always verifying your answer. These are fundamental concepts in algebra that will serve you well as you continue your mathematical journey.
Solving for variables is a core skill in algebra and beyond. It's used in countless applications in science, engineering, economics, and everyday life. By mastering these techniques, you're not just learning math; you're developing valuable problem-solving skills that will benefit you in many areas.
So, keep practicing, keep exploring, and never be afraid to ask questions. Math can be challenging, but it's also incredibly rewarding. With each problem you solve, you're building your understanding and expanding your abilities. You've got this! Now go out there and conquer those equations!