Hey guys! Let's dive into a common type of math problem that you'll likely encounter in algebra: solving for x when given a function h(x) and its value. In this case, we have the function h(x) = 4x - 5, and we want to find the value of x when h(x) = 7. Don't worry, it's not as intimidating as it might sound! We'll break it down step-by-step, so you'll be solving these problems like a pro in no time.
Understanding the Basics
Before we jump into the solution, let's make sure we're all on the same page with the basics. A function, like our h(x), is essentially a machine that takes an input (x in this case), performs some operations on it, and spits out an output. The expression 4x - 5 tells us exactly what operations the function performs: it multiplies the input x by 4 and then subtracts 5. When we say h(x) = 7, we're saying that the output of this "machine" is 7, and our goal is to figure out what input x would produce this output. Think of it like a puzzle – we know the answer, and we need to find the missing piece! This involves using algebraic manipulation to isolate x on one side of the equation. Algebraic manipulation is a fundamental skill in mathematics, allowing us to rearrange equations while maintaining their balance. It's like a mathematical dance, where we perform the same moves on both sides to keep everything in harmony. This includes operations such as addition, subtraction, multiplication, and division. The key is to perform the inverse operation to undo what's been done to x. For instance, if x has been multiplied by 4, we divide by 4. If 5 has been subtracted from x, we add 5. By systematically applying these inverse operations, we gradually peel away the layers surrounding x until it stands alone, revealing its value. This systematic approach not only solves the equation but also builds a deeper understanding of the relationships between variables and operations. So, let's roll up our sleeves and get started on solving our specific problem!
Step-by-Step Solution
Okay, let's get down to business and solve for x! Here's how we'll tackle this problem, step by step:
1. Set up the Equation
The first thing we need to do is write down the equation based on the information we have. We know that h(x) = 4x - 5 and h(x) = 7. So, we can set these two expressions equal to each other: 4x - 5 = 7. This equation is the heart of our problem, and everything we do from here will be aimed at solving it. Writing the equation correctly is crucial, as it forms the foundation for the rest of the solution. Any mistake here will lead to an incorrect answer, so double-check that you've transcribed the information accurately. Now that we have our equation, we're ready to start the process of isolating x. The strategy we'll use is to undo the operations that have been performed on x, one by one, until x is all by itself on one side of the equation. Think of it like unwrapping a gift – we need to carefully remove each layer to get to the prize inside! The order in which we undo these operations is important, and we'll generally follow the reverse order of operations (PEMDAS/BODMAS) to guide us. So, with our equation firmly in place, let's move on to the next step and start unwrapping!
2. Isolate the Term with x
Our goal is to get the x by itself, so let's start by getting rid of the -5 on the left side of the equation. To do this, we'll perform the inverse operation, which is adding 5. But remember, what we do to one side of the equation, we must do to the other side to keep things balanced. So, we'll add 5 to both sides: 4x - 5 + 5 = 7 + 5. This simplifies to 4x = 12. We've made significant progress here! We've successfully isolated the term containing x (which is 4x) on one side of the equation. This is a crucial step because it brings us closer to our ultimate goal of finding the value of x. The addition property of equality is the principle that allows us to add the same value to both sides of an equation without changing its solution. It's a fundamental tool in algebra, and we'll be using it repeatedly in this problem and in many others. Think of an equation as a balanced scale – if you add weight to one side, you must add the same weight to the other side to maintain the balance. By adding 5 to both sides, we've kept the equation balanced while moving us closer to isolating x. Now that we have 4x = 12, we're just one step away from solving for x. Let's move on to the final step and finish the job!
3. Solve for x
We're almost there! We have 4x = 12. Now, to get x by itself, we need to get rid of the 4 that's multiplying it. The inverse operation of multiplication is division, so we'll divide both sides of the equation by 4: (4x) / 4 = 12 / 4. This simplifies to x = 3. And there you have it! We've solved for x! The division property of equality, which we used in this step, is another fundamental principle in algebra. It states that you can divide both sides of an equation by the same non-zero number without changing the solution. Just like with addition, this property ensures that we maintain the balance of the equation while manipulating it to isolate the variable. Think of it as splitting a pizza equally – if you divide the pizza into a certain number of slices, each person gets the same amount regardless of how many people are sharing it. By dividing both sides by 4, we've effectively "unmultiplied" the x, leaving it all by itself and revealing its value. So, we've found that x = 3 is the solution to our problem. But before we celebrate too much, it's always a good idea to check our answer to make sure we haven't made any mistakes along the way.
Checking Our Answer
It's always a good idea to double-check your answer to make sure it's correct. To do this, we'll substitute our solution, x = 3, back into the original equation, h(x) = 4x - 5, and see if it gives us h(x) = 7. Let's plug it in: h(3) = 4(3) - 5. This simplifies to h(3) = 12 - 5, which further simplifies to h(3) = 7. Bingo! Our answer checks out! Substituting our solution back into the original equation is a crucial step in the problem-solving process. It's like a final exam for our work, ensuring that we haven't made any careless errors along the way. If our solution doesn't check out, it means we need to go back and review our steps to identify the mistake. Think of it as debugging a computer program – if the program doesn't run correctly, you need to trace back through the code to find the bug. By verifying our answer, we gain confidence in our solution and ensure that we're providing the correct answer. In this case, our solution x = 3 satisfies the equation h(x) = 7, so we can be sure that we've solved the problem correctly. Now that we've confirmed our solution, let's take a moment to recap the key steps we took to get there.
Conclusion
So, to solve for x when h(x) = 7 and h(x) = 4x - 5, we followed these steps:
- Set up the equation: 4x - 5 = 7
- Isolate the term with x: 4x = 12
- Solve for x: x = 3
- Checked our answer: h(3) = 7
Easy peasy, right? Solving for variables in equations is a fundamental skill in algebra, and it's one that you'll use again and again in more advanced math courses. By mastering these basic techniques, you're building a strong foundation for future success in mathematics. Remember, the key to success in math is practice, practice, practice! The more you work through problems, the more comfortable you'll become with the concepts and the more confident you'll feel in your ability to solve them. Think of learning math like learning a musical instrument – you need to put in the time and effort to develop your skills and build your "mathematical muscles." So, don't be afraid to tackle challenging problems and don't get discouraged if you make mistakes along the way. Every mistake is an opportunity to learn and grow. Keep practicing, and you'll be amazed at how far you can go! Now that we've successfully solved this problem, let's consider some additional tips and strategies that can help you tackle similar problems in the future.
Tips and Strategies for Solving Similar Problems
Here are a few tips and strategies that can help you solve similar problems:
- Always write down the equation first. This will help you visualize the problem and keep track of what you're doing. It's like creating a roadmap before embarking on a journey – it helps you stay on course and avoid getting lost!
- Remember to perform the same operation on both sides of the equation. This is crucial for maintaining balance and ensuring that you get the correct answer. Think of it as a seesaw – if you add weight to one side, you need to add the same weight to the other side to keep it level!
- Work backwards through the order of operations (PEMDAS/BODMAS) when isolating the variable. This will help you determine the correct order in which to undo the operations. It's like unwrapping a present – you need to remove the outer layers before you can get to the gift inside!
- Check your answer by substituting it back into the original equation. This will help you catch any mistakes and ensure that your solution is correct. It's like proofreading your writing – it helps you identify and correct any errors!
- Practice, practice, practice! The more you practice, the more comfortable you'll become with solving equations. Think of it as training for a marathon – you need to put in the miles to build your endurance and reach the finish line!
By following these tips and strategies, you'll be well-equipped to tackle a wide range of algebraic problems. Solving equations is a fundamental skill that opens the door to more advanced mathematical concepts, so it's worth investing the time and effort to master it. Remember, mathematics is a journey, not a destination. Enjoy the process of learning and problem-solving, and you'll be amazed at what you can achieve!
I hope this guide has been helpful! Keep practicing, and you'll become a master at solving for x in no time!