Solving Linear Equations: A Step-by-Step Guide

Let's dive into solving the linear equation: 27p - 3 = 1 + 216 - 9p - 2 - 2 + 1 - 4p. Don't worry, guys, it looks more complicated than it actually is! We'll break it down step by step so you can totally master it. Linear equations are the foundation of algebra, and understanding how to solve them opens up a whole new world of mathematical possibilities.

Understanding the Basics

Before we jump into solving, let's make sure we're all on the same page. A linear equation is an equation where the highest power of the variable (in this case, 'p') is 1. Our goal is to isolate 'p' on one side of the equation to find its value. We do this by performing the same operations on both sides of the equation to maintain balance. Think of it like a seesaw – whatever you do to one side, you have to do to the other to keep it level. Remember that linear equations are like the bread and butter of basic algebra, so getting comfortable with them is super important.

When we talk about 'solving,' we're essentially trying to find the value of the variable that makes the equation true. This value is often called the 'solution' or 'root' of the equation. For instance, if we find that p = 5, it means that if we substitute 5 for 'p' in the original equation, both sides of the equation will be equal. It's like finding the missing piece of a puzzle. So, bear with me as we go through the steps; you'll be solving these like a pro in no time!

Step 1: Simplify Both Sides of the Equation

The first thing we need to do is simplify both sides of the equation. This means combining any like terms. On the left side, we have 27p - 3, which is already simplified. On the right side, we have 1 + 216 - 9p - 2 - 2 + 1 - 4p. Let's combine the constant terms (the numbers without 'p') and the 'p' terms separately. For the constant terms: 1 + 216 - 2 - 2 + 1 = 214. For the 'p' terms: -9p - 4p = -13p. So, the right side simplifies to 214 - 13p.

Now our equation looks like this: 27p - 3 = 214 - 13p. See? Much cleaner already! Simplifying is a crucial step because it makes the equation easier to work with. It's like decluttering your workspace before starting a big project – it helps you focus and reduces the chance of making mistakes. This step involves basic arithmetic, so make sure to double-check your calculations to avoid any errors. Trust me, a small mistake here can throw off your entire solution, and nobody wants that!

Step 2: Move the 'p' Terms to One Side

Next, we want to get all the 'p' terms on one side of the equation. It doesn't matter which side you choose, but it's often easier to move the terms so that the coefficient of 'p' remains positive. In this case, we can add 13p to both sides of the equation. This will eliminate the '-13p' term on the right side and move it to the left side. So, we have: 27p - 3 + 13p = 214 - 13p + 13p. Simplifying, we get 40p - 3 = 214.

The key here is to remember that whatever you do to one side, you must do to the other. Adding 13p to both sides keeps the equation balanced. It's like adding the same weight to both sides of a scale – it stays level. Moving the 'p' terms to one side is a fundamental step in solving linear equations. It isolates the variable, bringing us closer to finding its value. Also, keep in mind that you could have chosen to move the 'p' terms to the right side instead, but this would have resulted in a negative coefficient for 'p', which might be a bit trickier to work with. It's all about making the equation as manageable as possible!

Step 3: Move the Constant Terms to the Other Side

Now we want to isolate the 'p' term further by moving all the constant terms to the other side of the equation. We can do this by adding 3 to both sides of the equation. This will eliminate the '-3' term on the left side and move it to the right side. So, we have: 40p - 3 + 3 = 214 + 3. Simplifying, we get 40p = 217.

Just like before, it's crucial to maintain balance by performing the same operation on both sides. Adding 3 to both sides ensures that the equation remains true. This step brings us even closer to isolating 'p'. Remember, our goal is to get 'p' all by itself on one side of the equation. Moving the constant terms is a key part of this process. It's like separating the ingredients in a recipe – you need to isolate the one you're most interested in before you can work with it. At this point, we're almost there, guys! Just one more step to go!

Step 4: Solve for 'p'

Finally, to solve for 'p', we need to divide both sides of the equation by the coefficient of 'p', which is 40. So, we have: 40p / 40 = 217 / 40. Simplifying, we get p = 217/40. This is our solution! We can leave it as an improper fraction or convert it to a decimal, which is approximately 5.425.

Congratulations, you've solved the equation! Dividing both sides by the coefficient of 'p' is the final step in isolating the variable. It's like the last piece of the puzzle falling into place. Once you've done this, you've found the value of 'p' that makes the equation true. In this case, p = 217/40 or approximately 5.425. You can always check your answer by substituting it back into the original equation to see if both sides are equal. This is a great way to ensure that you haven't made any mistakes along the way. Solving linear equations might seem daunting at first, but with practice, it becomes second nature. Keep practicing, and you'll be a pro in no time!

Conclusion

So, to recap, solving the equation 27p - 3 = 1 + 216 - 9p - 2 - 2 + 1 - 4p involves simplifying both sides, moving the 'p' terms to one side, moving the constant terms to the other side, and finally, solving for 'p'. Remember to always perform the same operations on both sides of the equation to maintain balance. With practice, you'll be able to solve linear equations like a boss! Keep up the great work, and happy solving! Linear equations are fundamental, and mastering them will pave the way for more advanced topics in algebra and beyond. So, embrace the challenge, and never stop learning!