Hey guys! Today, we're diving into solving a linear equation and, just as importantly, checking our work to make sure we've got the right answer. We'll break down each step, making it super clear and easy to follow. The equation we're tackling is: -(3/4)h + 23 = 11. Stick around, and you'll be solving equations like a pro in no time!
Step-by-Step Solution
1. Isolate the Term with the Variable
Okay, so the first thing we want to do is get the term with our variable, 'h' in this case, all by itself on one side of the equation. To do that, we need to get rid of the '+ 23' that's hanging out on the left side. The way we do this is by performing the inverse operation. Since we're adding 23, we're going to subtract 23 from both sides of the equation. Remember, whatever you do to one side, you've gotta do to the other to keep things balanced!
So, our equation becomes:
-(3/4)h + 23 - 23 = 11 - 23
This simplifies to:
-(3/4)h = -12
Great! We've isolated the term with 'h'. Now, let's move on to the next step.
2. Get Rid of the Fraction
Fractions can sometimes look a little intimidating, but don't worry, we've got this! We need to get 'h' completely by itself, and right now, it's being multiplied by -3/4. To undo this multiplication, we're going to multiply both sides of the equation by the reciprocal of -3/4. The reciprocal is just the fraction flipped over, so the reciprocal of -3/4 is -4/3. Multiplying by the reciprocal is the same as dividing, but it's often easier to think of it this way.
Here's how it looks:
(-4/3) * (-(3/4)h) = (-4/3) * (-12)
On the left side, the -4/3 and -3/4 cancel each other out (they multiply to 1), leaving us with just 'h'. On the right side, we've got (-4/3) * (-12). Remember, a negative times a negative is a positive. We can think of -12 as -12/1, so we're multiplying two fractions: (-4/3) * (-12/1). Multiply the numerators (-4 * -12 = 48) and multiply the denominators (3 * 1 = 3). So we have 48/3, which simplifies to 16.
So, we now have:
h = 16
Awesome! We've solved for 'h'. But before we do a victory dance, we need to make sure our answer is correct.
Checking the Solution
This is a crucial step, guys! Checking your solution is like having a superpower in math. It lets you know for sure if you've nailed it or if you need to go back and take another look. To check our solution, we're going to plug our value for 'h' (which is 16) back into the original equation and see if it makes the equation true.
Our original equation was:
-(3/4)h + 23 = 11
Substitute h = 16:
-(3/4)(16) + 23 = 11
Now, let's simplify. First, we'll multiply -(3/4) by 16. We can think of 16 as 16/1, so we have -(3/4) * (16/1). Multiply the numerators (-3 * 16 = -48) and multiply the denominators (4 * 1 = 4). So we have -48/4, which simplifies to -12.
Our equation now looks like this:
-12 + 23 = 11
Is this true? Yep! -12 + 23 does indeed equal 11.
Since plugging in h = 16 makes the equation true, we know that 16 is the correct solution.
Why Checking Your Work is So Important
Okay, guys, let's talk about why this checking step isn't just some extra thing your math teacher makes you do. It's actually a super powerful tool for a few reasons:
- Catches Mistakes: We all make mistakes, especially when we're working through multi-step problems. A simple sign error or a miscalculation can throw off the whole answer. Checking your work lets you catch those errors before you turn in your work or move on to the next problem. Think of it as your personal error-detection system!
- Builds Confidence: When you check your work and it works out, you can feel confident that you've got the right answer. This is a great feeling, and it helps you build confidence in your math skills overall. It's like getting a gold star from yourself!
- Deepens Understanding: The act of plugging your solution back into the original equation and simplifying actually helps you understand the problem better. You're reinforcing the concepts and seeing how everything fits together. It's like a mini-review session built right into the problem-solving process.
- Prevents Future Errors: When you identify a mistake during the checking process, you can learn from it. Maybe you consistently forget to distribute a negative sign, or maybe you struggle with fraction multiplication. By catching these errors, you can be more mindful of them in the future and avoid making the same mistakes again. It's like leveling up your math skills!
So, seriously, don't skip the checking step. It's not just busywork; it's a valuable part of the problem-solving process that can save you time, frustration, and points on tests!
Key Concepts Review
Before we wrap up, let's quickly review the key concepts we used to solve this equation:
- Inverse Operations: We used inverse operations (subtraction to undo addition, multiplication to undo division) to isolate the variable. This is a fundamental concept in solving equations.
- Reciprocals: We multiplied by the reciprocal of a fraction to get rid of it. Remember, the reciprocal is just the fraction flipped over.
- Maintaining Balance: Whatever operation we perform on one side of the equation, we must perform on the other side to keep the equation balanced.
- Substitution: We substituted our solution back into the original equation to check our work. Substitution is a powerful tool in algebra and beyond.
Real-World Applications
Solving linear equations might seem like an abstract math concept, but it actually has tons of real-world applications. Here are a few examples:
- Budgeting: Let's say you have a certain amount of money to spend on groceries, and you know the price of some of the items you need. You can use a linear equation to figure out how much you can spend on other items.
- Calculating Distances: If you know the speed you're traveling and the time you've been traveling, you can use a linear equation to calculate the distance you've covered.
- Mixing Solutions: In chemistry and other fields, you might need to mix solutions of different concentrations to get a desired concentration. Linear equations can help you figure out how much of each solution to use.
- Finance: Calculating loan payments, interest rates, and investment returns often involves linear equations.
So, the skills you're learning in algebra are actually super useful in many different areas of life!
Practice Makes Perfect
Alright, guys, we've solved one equation together, but the best way to really master this skill is to practice. Try solving some more equations on your own. You can find practice problems in your textbook, online, or even make up your own! Remember to always check your solutions to make sure you're on the right track.
The more you practice, the more comfortable and confident you'll become with solving equations. And who knows, maybe you'll even start to enjoy it!
Conclusion
So, there you have it! We've successfully solved the equation -(3/4)h + 23 = 11 and verified that our solution, h = 16, is correct. We've also discussed why checking your work is so important and touched on some real-world applications of linear equations. Keep practicing, and you'll be solving equations like a math whiz in no time! You've got this!
Remember, math isn't just about getting the right answer; it's about understanding the process and building your problem-solving skills. So, keep exploring, keep questioning, and keep learning!