Hey guys! Today, we're diving into a fun math problem. We've got an equation that Noah started to solve, and we're going to help him finish it. The equation is -4.6p - 6.3p + 3.9 = -9.18. Noah's already made a great start by combining like terms, which is always a smart move. He simplified the equation to -10.9p + 3.9 = -9.18. Now, our mission is to figure out the next steps and find the value of p. Let's break it down together!
Understanding the Problem
Before we jump into solving, let's make sure we understand what's going on. We have a linear equation, which means it involves a variable (p in this case) raised to the power of 1. Our goal is to isolate p on one side of the equation so we can see what it equals. To do this, we'll use inverse operations – basically, doing the opposite of what's being done to p. Think of it like unwrapping a present; we need to undo each layer to get to the gift inside. In this case, the gift is the value of p. Remember, the key to solving equations is to keep both sides balanced. Whatever operation we perform on one side, we must perform on the other side to maintain the equality. Math is all about balance, guys!
Step 1: Isolate the Term with 'p'
The first thing we want to do is get the term with p (-10.9p) by itself on one side of the equation. Currently, we have a +3.9 hanging out with it. To get rid of it, we need to perform the inverse operation. Since 3.9 is being added, we'll subtract 3.9 from both sides of the equation. This is a crucial step, so let's write it out:
-10.9p + 3.9 - 3.9 = -9.18 - 3.9
On the left side, the +3.9 and -3.9 cancel each other out, leaving us with just -10.9p. On the right side, we have -9.18 - 3.9, which equals -13.08. So, our equation now looks like this:
-10.9p = -13.08
We're one step closer to solving for p! See how we're methodically peeling away the layers? Each step brings us closer to our goal. Remember, staying organized and writing out each step clearly helps prevent errors and makes the process much smoother. Math can be like a puzzle, and each step is a piece fitting into place.
Step 2: Solve for 'p'
Now we have -10.9p = -13.08. The next step is to isolate p completely. Currently, p is being multiplied by -10.9. To undo this multiplication, we'll use the inverse operation: division. We'll divide both sides of the equation by -10.9. This is where paying attention to signs is super important. A negative divided by a negative will give us a positive, which is a little sneak peek at what our answer will look like.
Here's the equation with the division:
-10.9p / -10.9 = -13.08 / -10.9
On the left side, -10.9 divided by -10.9 is 1, so we're left with just p. On the right side, -13.08 divided by -10.9 is approximately 1.2. Let's write that down:
p = 1.2
And there we have it! We've solved for p. It's like finding the treasure at the end of a mathematical journey. But before we celebrate, let's do one more thing to make sure we've got it right.
Step 3: Check Your Solution
It's always a good idea to check your answer, guys. This helps catch any mistakes and gives you confidence that you've solved the problem correctly. To check our solution, we'll substitute p = 1.2 back into the original equation: -4.6p - 6.3p + 3.9 = -9.18.
Let's plug it in:
-4.6(1.2) - 6.3(1.2) + 3.9 = -9.18
Now, we perform the multiplications:
-5.52 - 7.56 + 3.9 = -9.18
Next, we add and subtract:
-13.08 + 3.9 = -9.18
-9.18 = -9.18
Woohoo! The equation holds true. This means our solution, p = 1.2, is correct. Checking your work is like having a safety net; it's there to catch you if you stumble. It's an essential step in the problem-solving process, especially in math. Now we can confidently say we've conquered this equation!
Final Answer
So, the solution to the equation -4.6p - 6.3p + 3.9 = -9.18 is:
p = 1.2
Great job, guys! We worked through this problem step by step, and we found the answer. Remember, solving equations is all about understanding the process, using inverse operations, and keeping things balanced. And don't forget to check your work! With practice, you'll become equation-solving pros. Math can be challenging, but it's also super rewarding when you figure things out. Keep practicing, keep asking questions, and keep having fun with math!
Key Takeaways
- Combine Like Terms First: This simplifies the equation and makes it easier to work with. It's like decluttering your workspace before starting a project.
- Use Inverse Operations: To isolate the variable, perform the opposite operation (addition/subtraction, multiplication/division). Think of it as undoing the steps that were done to the variable.
- Keep the Equation Balanced: Whatever you do to one side, you must do to the other. This is the golden rule of equation solving.
- Check Your Solution: Plug your answer back into the original equation to make sure it works. This is your safety net and ensures accuracy.
These key takeaways are like the essential tools in your math toolbox. Keep them handy, and you'll be well-equipped to tackle any equation that comes your way.
Practice Makes Perfect
Solving equations is like learning a new skill; it takes practice. The more you practice, the more comfortable and confident you'll become. So, don't be afraid to try more problems. You can find practice problems in textbooks, online resources, or even create your own. The key is to keep challenging yourself and to learn from your mistakes. Mistakes are just opportunities to learn and grow, guys!
Conclusion
We've successfully solved the equation -4.6p - 6.3p + 3.9 = -9.18, and we've learned some valuable strategies along the way. Remember, math is not just about finding the right answer; it's about understanding the process and developing problem-solving skills. So, keep exploring, keep learning, and keep enjoying the world of mathematics! You've got this!