Solve For P In The Equation 24p + 12 - 18p = 10 + 2p - 6 A Step-by-Step Guide

Hey guys! Today, we're diving deep into the world of linear equations to solve for the elusive value of p in the equation 24p + 12 - 18p = 10 + 2p - 6. Don't worry if it looks a bit intimidating at first glance; we're going to break it down step by step, making it super easy to understand. Think of it like solving a puzzle – each step brings us closer to the final answer. So, grab your pencils, notebooks, and let's get started on this mathematical adventure!

Understanding Linear Equations

First, let's get a grip on what we're dealing with. Linear equations, at their core, are algebraic equations where the highest power of the variable (in our case, p) is 1. These equations represent a straight line when graphed, hence the name "linear." Now, why are linear equations so important, you might ask? Well, they're fundamental in countless real-world applications, from calculating distances and speeds to predicting trends in economics and science. They're the building blocks of more complex mathematical models, so mastering them is crucial.

In our particular equation, 24p + 12 - 18p = 10 + 2p - 6, we have p terms, constant terms, and an equality sign that tells us the expression on the left side is equal to the expression on the right side. Our mission is to isolate p on one side of the equation to find its value. Remember those algebra classes where you felt like you were decoding a secret message? This is exactly that, but we're the codebreakers!

To start, we need to simplify both sides of the equation separately. This involves combining like terms – terms that have the same variable raised to the same power (or no variable at all, in the case of constants). On the left side, we can combine 24p and -18p, and on the right side, we can combine the constant terms 10 and -6. This is like sorting your socks – you put all the same types together to make things easier to manage. Once we've simplified each side, the equation will look much cleaner and we'll be one step closer to finding p.

Simplifying the Equation: A Step-by-Step Guide

The crucial first step in solving any linear equation is simplification. In our equation, 24p + 12 - 18p = 10 + 2p - 6, we have some terms that can be combined to make things easier. Think of it as tidying up before you start a big project – a clean workspace helps you think clearly. On the left side, we have 24p and -18p. These are like terms because they both contain the variable p raised to the power of 1. To combine them, we simply add their coefficients (the numbers in front of the p): 24 - 18 = 6. So, 24p - 18p becomes 6p. This is like saying you have 24 apples and you give away 18; you're left with 6 apples.

Next, let's look at the right side of the equation: 10 + 2p - 6. Here, we have constant terms 10 and -6. These are just plain numbers, so we can add or subtract them directly. 10 - 6 equals 4. So, the right side of the equation simplifies to 4 + 2p. We haven't touched the 2p term yet because it doesn't have any like terms on this side of the equation.

After this initial simplification, our equation now looks like this: 6p + 12 = 4 + 2*p. See how much cleaner it is? We've taken a somewhat messy equation and transformed it into something much more manageable. This is a key strategy in algebra – breaking down complex problems into simpler steps. Now, we're ready to move on to the next phase: isolating the p terms on one side of the equation. This is like separating the ingredients you need for a recipe – you want them all in one place so you can work with them efficiently.

Isolating the Variable: Getting p on Its Own

Now that we've simplified both sides of our equation, the next strategic move is to isolate the variable p. This means we want all the terms containing p on one side of the equation and all the constant terms on the other side. It's like sorting your laundry – you put all the shirts in one pile and all the pants in another. In our equation, 6p + 12 = 4 + 2*p, we need to get the p terms together and the numbers together. Remember, the golden rule of equations is that whatever you do to one side, you must do to the other to maintain the balance.

Let's start by moving the 2p term from the right side to the left side. To do this, we subtract 2p from both sides of the equation. Why subtract? Because we want to "cancel out" the 2p on the right side. This gives us: 6p - 2p + 12 = 4 + 2p - 2p. On the right side, 2p - 2p becomes 0, so we're left with just 4. On the left side, 6p - 2p combines to 4p. So, our equation now looks like this: 4p + 12 = 4. We've successfully moved all the p terms to the left side!

Next, we need to move the constant term, 12, from the left side to the right side. We do this by subtracting 12 from both sides of the equation. This gives us: 4p + 12 - 12 = 4 - 12. On the left side, 12 - 12 becomes 0, leaving us with just 4p. On the right side, 4 - 12 equals -8. So, our equation is now: 4p = -8. We're almost there! We've managed to isolate the p term on one side of the equation. The final step is to get p completely on its own, without any coefficient (the number in front of it).

Solving for p: The Final Step

We've reached the home stretch! Our equation is now simplified to 4p = -8. The last hurdle is to get p completely isolated, meaning we need to get rid of the coefficient 4 that's multiplying p. Remember, the key to solving equations is to do the opposite operation to "undo" what's happening to the variable. In this case, p is being multiplied by 4, so to undo that, we need to divide.

We'll divide both sides of the equation by 4. This gives us: (4p) / 4 = -8 / 4. On the left side, the 4 in the numerator and the 4 in the denominator cancel each other out, leaving us with just p. On the right side, -8 divided by 4 equals -2. So, we have our answer: p = -2! We've successfully solved the equation and found the value of p.

Think of this step like unwrapping a gift – you're peeling away the layers to reveal the treasure inside, which in this case is the value of p. It's a satisfying feeling to reach the end of a problem and know you've cracked the code. But, just like a good scientist, we shouldn't just accept our answer without checking it. Let's move on to verifying our solution to make sure everything adds up.

Verifying the Solution: Double-Checking Our Work

Alright, guys, we've got a potential answer, p = -2, but before we celebrate, it's crucial to verify our solution. Think of this as the quality control step – we want to make sure our answer is correct and that we haven't made any sneaky errors along the way. Verifying our solution is like proofreading a piece of writing; it helps catch any mistakes we might have missed.

To verify, we'll take our value of p, which is -2, and plug it back into the original equation: 24p + 12 - 18p = 10 + 2p - 6. We're essentially asking, "Does this value of p make the equation true?" If it does, then we know we've solved it correctly. If not, we need to go back and look for any mistakes in our steps.

Let's substitute -2 for p in the equation: 24*(-2) + 12 - 18*(-2) = 10 + 2*(-2) - 6. Now, we need to perform the arithmetic operations, following the order of operations (PEMDAS/BODMAS). First, we'll do the multiplications: 24*(-2) = -48, -18*(-2) = 36, and 2*(-2) = -4. So, our equation becomes: -48 + 12 + 36 = 10 - 4 - 6.

Next, we'll do the additions and subtractions on both sides of the equation. On the left side, -48 + 12 + 36 = 0. On the right side, 10 - 4 - 6 = 0. So, we have 0 = 0. This is a true statement! It means that when we substitute p = -2 into the original equation, both sides of the equation are equal. This confirms that our solution is correct.

We've successfully verified our solution! It's like getting a gold star on a test – it's a great feeling to know you've done it right. This step is a critical part of problem-solving in mathematics and in life. Always double-check your work, whether you're solving an equation or making an important decision. Now that we're confident in our answer, let's summarize what we've learned.

Conclusion: The Value of p Revealed

So, guys, we've journeyed through the world of linear equations, tackled the equation 24p + 12 - 18p = 10 + 2p - 6, and emerged victorious! We've systematically simplified the equation, isolated the variable p, and finally, discovered its value: p = -2. This wasn't just about finding a number; it was about understanding the process, the logic, and the art of problem-solving.

We started by understanding the basic principles of linear equations, recognizing their importance in mathematics and real-world applications. We then broke down the problem into manageable steps: simplifying both sides of the equation, isolating the p terms, and solving for p. Each step was like a piece of a puzzle, and when we put them all together, we revealed the solution. And remember, we didn't just stop at finding the answer; we verified it to ensure our solution was correct. This is a testament to the importance of precision and accuracy in mathematics.

This exercise wasn't just about math; it was about developing critical thinking skills, logical reasoning, and the ability to approach complex problems with confidence. These are skills that will serve you well in all areas of life, whether you're solving a mathematical equation or making a decision about your future. So, the next time you encounter a challenging problem, remember the steps we took today, and you'll be well-equipped to tackle it head-on. Keep practicing, keep exploring, and keep unlocking the mysteries of mathematics!