Solve For H: Step-by-Step Equation Guide
Hey guys! Today, we're diving into a fun little algebraic puzzle. We've got the equation f = (1/7)(g + h - k), and our mission, should we choose to accept it, is to isolate h. This means we need to rearrange the equation so that h is all by itself on one side. Think of it like untangling a string of holiday lights – a little bit of patience and the right moves will get you there! This kind of problem is super common in math and science, where you often need to rearrange formulas to find a specific variable. So, let's roll up our sleeves and get to it! We'll break it down step by step, so it's crystal clear how to solve for h. Whether you're a math whiz or just starting out, this guide will help you master this type of equation manipulation. We'll cover the basic principles of algebraic manipulation, emphasizing how to maintain the balance of the equation while isolating the variable of interest. Understanding these techniques is crucial not just for solving this particular problem but for tackling a wide range of mathematical challenges. Stick with us, and you'll be a pro at solving for h in no time!
Why is solving for a specific variable so important? Well, in many real-world scenarios, we have formulas that describe relationships between different quantities. But sometimes, we know the values of some of those quantities and need to find the value of another. That's where rearranging equations comes in handy. For example, in physics, you might use a formula to calculate the distance traveled by an object. But if you know the distance and the time, you might need to rearrange the formula to find the object's speed. Similarly, in economics, you might use a formula to calculate the profit of a business. But if you know the profit and the revenue, you might need to rearrange the formula to find the costs. The ability to manipulate equations and solve for specific variables is a fundamental skill that opens doors to problem-solving in various fields. It's not just about memorizing formulas; it's about understanding how to manipulate them to extract the information you need.
The first thing we want to do is ditch that fraction, right? Fractions can sometimes make things look a little messier than they need to be. In our equation, f = (1/7)(g + h - k), we've got that 1/7 hanging out front. To get rid of it, we can multiply both sides of the equation by 7. Remember, in algebra, whatever we do to one side, we gotta do to the other to keep things balanced. It's like a see-saw – if you add weight to one side, you need to add the same weight to the other to keep it level. So, let's multiply both sides by 7:
7 * f = 7 * (1/7)(g + h - k)
On the right side, the 7 and the 1/7 cancel each other out, which is exactly what we wanted! This leaves us with:
7f = g + h - k
Why does this work? Multiplying by the reciprocal (in this case, 7 is the reciprocal of 1/7) is a fundamental technique for clearing fractions in equations. It simplifies the equation and makes it easier to work with. Think of it as undoing the division that the fraction represents. By multiplying, we're essentially reversing the operation and isolating the terms we want to deal with. This step is crucial because it sets the stage for further manipulations to isolate h. Without clearing the fraction, we'd have to deal with it throughout the rest of the solution, making the process more complex. By eliminating it early on, we streamline the process and reduce the chances of making errors. So, remember this trick – it's a lifesaver when you're dealing with equations involving fractions!
Okay, we've cleared the fraction and now our equation looks much cleaner: 7f = g + h - k. Our next goal is to get the term with h (which is just h itself in this case) all by its lonesome on one side of the equation. To do this, we need to get rid of the g and the -k that are hanging out with it. We can do this by performing the opposite operations on both sides of the equation. Remember, it's all about keeping that equation balanced! g is being added, so we'll subtract g from both sides. And -k is being subtracted, so we'll add k to both sides. Let's do it:
7f - g + k = g + h - k - g + k
On the right side, the g and -g cancel each other out, and the -k and +k also cancel each other out. This leaves us with:
7f - g + k = h
What's the logic behind this step? The key principle here is the concept of inverse operations. Addition and subtraction are inverse operations, meaning they undo each other. By subtracting g from both sides, we effectively moved it from the right side to the left side of the equation. Similarly, by adding k to both sides, we moved it as well. This process is crucial for isolating the variable we're solving for. Think of it as peeling away the layers surrounding h until it stands alone. Each operation we perform brings us closer to our goal. By strategically using inverse operations, we can systematically eliminate terms and simplify the equation until we have h isolated. This step demonstrates the power of algebraic manipulation in rearranging equations to reveal the relationship between variables.
Alright, check it out! We've done it! We've successfully isolated h. Our equation now reads:
7f - g + k = h
This is the same as:
h = 7f - g + k
We just flipped the sides for clarity, but they mean exactly the same thing. This is our final answer! We've solved for h in terms of f, g, and k. Pat yourselves on the back, guys! You've tackled an algebraic equation and come out on top. This means that if we know the values of f, g, and k, we can easily plug them into this equation and calculate the value of h. That's the power of solving for a variable – it gives us a direct way to find its value based on the values of other variables in the equation.
Why is this the final answer? Because we've achieved our initial goal: we've expressed h in terms of the other variables in the equation. There are no more operations we can perform to further isolate h. The equation is in its simplest form, where h is equal to an expression involving f, g, and k. This form is incredibly useful because it allows us to directly calculate h if we know the values of the other variables. In many real-world applications, this is exactly what we need to do. We might have a formula that relates several quantities, and we want to find the value of one quantity based on the values of the others. Solving for a specific variable, like we did for h, gives us the tool to do just that. So, congratulations on reaching the final answer! You've demonstrated a key skill in algebra that will serve you well in many different contexts.
So there you have it! We've successfully solved for h in the equation f = (1/7)(g + h - k). We went step-by-step, first clearing the fraction by multiplying both sides by 7, then isolating the term with h by using inverse operations. Remember, the key to solving these types of equations is to keep everything balanced and to perform the same operations on both sides. Solving for variables is a fundamental skill in algebra and is used in many different fields, from science to engineering to economics. The ability to manipulate equations and isolate variables allows us to understand relationships between quantities and to make predictions based on those relationships. It's a powerful tool that empowers us to solve problems and make informed decisions.
What's the takeaway? The process of solving for a variable is not just about getting the right answer; it's about understanding the underlying principles of algebraic manipulation. It's about recognizing the importance of balance in equations and the power of inverse operations. By mastering these concepts, you'll be well-equipped to tackle more complex mathematical challenges. Practice is key, so don't be afraid to try solving similar equations on your own. The more you practice, the more comfortable you'll become with the process, and the more confident you'll be in your ability to solve for any variable in any equation. So, keep practicing, keep learning, and keep exploring the fascinating world of algebra!