Solving 3(p-5)+4p=p+9: A Step-by-Step Guide

Hey guys! Let's dive into solving this equation together. It might look a little intimidating at first, but trust me, we'll break it down step-by-step so it's super easy to understand. We're tackling the equation $3(p-5)+4p=p+9$, and the goal is to find out what value of 'p' makes this equation true. So grab your pencils, paper, and let’s get started!

Unpacking the Equation: A Journey Through the Steps

1. Distribute Like a Boss: Unleashing the Power of Multiplication

Okay, the very first thing we need to do is handle those parentheses. We've got $3(p-5)$ sitting there, and that's a clear signal to distribute. Distribution is just a fancy way of saying we're going to multiply the 3 by everything inside the parentheses. So, we multiply 3 by 'p', which gives us $3p$, and then we multiply 3 by -5, which gives us $-15$. Remember to keep that negative sign! Our equation now looks like this: $3p - 15 + 4p = p + 9$. See? We've already made progress, and it's not so scary anymore. The key here is to take your time and make sure you're multiplying correctly. A little mistake in distribution can throw off the whole answer, so double-check your work. Think of distribution as unlocking a door – once you do it right, you can move on to the next part of the problem. It's like leveling up in a game! We've successfully distributed, and now we're ready to combine those like terms and make the equation even simpler. We are one step closer to finding the value of 'p' and conquering this mathematical challenge.

2. Combining Like Terms: Gathering the Troops

Alright, now that we've distributed, it's time to gather our like terms. This is like sorting your socks after laundry day – you want to put all the matching ones together. In our equation, $3p - 15 + 4p = p + 9$, we have two terms with 'p' on the left side: $3p$ and $4p$. These are our like terms, and we can combine them by simply adding their coefficients (the numbers in front of the 'p'). So, $3p + 4p$ equals $7p$. Now our equation looks even cleaner: $7p - 15 = p + 9$. See how much simpler it's becoming? Combining like terms is a crucial step in solving equations. It helps us consolidate the equation and makes it easier to isolate our variable, which is 'p' in this case. We're essentially tidying up the equation, making it more manageable. It's like decluttering your room – once everything is organized, it's much easier to find what you're looking for. In this case, we're looking for the value of 'p', and by combining like terms, we've made the path to finding it much clearer. So, we've successfully combined our 'p' terms, and we're ready to move on to the next step: isolating 'p' on one side of the equation. We're on a roll, guys!

3. Isolating the Variable: Getting 'p' All Alone

Okay, the name of the game now is to get 'p' all by itself on one side of the equation. We want to isolate that 'p' like it's the last piece of pizza, and we're really hungry. Right now, we have $7p - 15 = p + 9$. Notice that 'p' appears on both sides of the equation. Our goal is to get all the 'p' terms on one side and all the constant terms (the numbers without 'p') on the other side. Let's start by getting rid of the 'p' on the right side. To do this, we'll subtract 'p' from both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep things balanced. So, $7p - p - 15 = p - p + 9$. This simplifies to $6p - 15 = 9$. We've successfully moved the 'p' terms to the left side, and now we just have a constant term (+9) on the right side. Now, let's get rid of that $-15$ on the left side. To do this, we'll add 15 to both sides of the equation: $6p - 15 + 15 = 9 + 15$. This simplifies to $6p = 24$. We're so close! We've managed to isolate the 'p' term on the left side, and we have a constant term on the right side. Now, there's just one more little step to get 'p' completely alone. We're almost there, guys! Isolating the variable is a fundamental skill in algebra, and we're mastering it step by step.

4. Solving for 'p': The Grand Finale

Drumroll, please! We've reached the final step in our equation-solving adventure. We're at $6p = 24$, and all that's left to do is figure out what 'p' is. Remember that $6p$ means 6 multiplied by 'p'. To undo this multiplication and get 'p' by itself, we need to do the opposite operation: division. We're going to divide both sides of the equation by 6. So, $(6p) / 6 = 24 / 6$. On the left side, the 6s cancel out, leaving us with just 'p'. On the right side, 24 divided by 6 is 4. So, we have our answer: $p = 4$. Hooray! We've successfully solved the equation. We found the value of 'p' that makes the equation $3(p-5) + 4p = p + 9$ true. Solving for the variable is the ultimate goal in these types of problems. It's like reaching the summit of a mountain after a long climb – the view from the top is totally worth it. In this case, the "view" is the satisfaction of knowing we've conquered the equation and found the value of 'p'. We took it step by step, distributed, combined like terms, isolated the variable, and finally solved for 'p'. And now, we can confidently say that $p = 4$ is the solution. Great job, team!

The Victory Lap: Verifying Our Solution

Now, just to be extra sure we nailed it, let's do a quick check. We're going to plug our solution, $p = 4$, back into the original equation, $3(p-5) + 4p = p + 9$, and see if both sides of the equation are equal. This is like double-checking your work on a test – it's always a good idea to make sure you didn't make any silly mistakes. So, let's substitute $p = 4$ into the equation: $3(4-5) + 4(4) = 4 + 9$. Now, we simplify. First, we deal with the parentheses: $3(-1) + 4(4) = 4 + 9$. Next, we multiply: $-3 + 16 = 4 + 9$. Finally, we add: $13 = 13$. Boom! The left side equals the right side. This means our solution, $p = 4$, is correct. We did it! Verifying our solution is a crucial step in the problem-solving process. It gives us confidence in our answer and helps us catch any errors we might have made along the way. It's like having a safety net – it's there to protect us from falling. In this case, our safety net confirmed that we got the right answer. So, we can proudly say that we've not only solved the equation but also verified our solution. Give yourselves a pat on the back, guys! You've successfully navigated the world of equation-solving.

Wrapping Up: Equation-Solving Superstars

So, there you have it, guys! We've successfully solved the equation $3(p-5) + 4p = p + 9$, and we found that $p = 4$. We walked through each step together, from distributing to combining like terms, isolating the variable, and finally solving for 'p'. We even did a quick check to make sure our answer was correct. Solving equations might seem tricky at first, but with practice and a step-by-step approach, you can conquer any equation that comes your way. Remember, the key is to take your time, stay organized, and double-check your work. And most importantly, don't be afraid to ask for help if you get stuck. Math is a team sport, and we're all in this together. You've got this! Keep practicing, keep learning, and keep solving. You're all equation-solving superstars! And remember, every problem you solve makes you a little bit stronger and a little bit more confident. So, keep challenging yourself, and you'll be amazed at what you can accomplish. Now, go forth and conquer more equations! You've got the skills, you've got the knowledge, and you've got the determination. The world of math is waiting for you!