Solving 3p - 7 + P = 13 First Step Explained With Examples

Hey guys! Today, we're diving into a super common type of math problem: solving equations. Specifically, we're going to break down the equation 3p - 7 + p = 13. Don't worry if it looks intimidating at first. We'll take it one step at a time, just like you would in a real math class or while studying for an exam. Our mission is to figure out the resulting equation after the very first step in solving it. This is a foundational skill in algebra, and mastering it will make more complex problems seem way easier. So, grab your mental math tools, and let's get started!

Understanding the Initial Equation

Before we jump into solving, let's make sure we fully understand what the equation 3p - 7 + p = 13 is telling us. In algebra, an equation is like a balanced scale. The equals sign (=) means that whatever is on the left side has the exact same value as whatever is on the right side. Our equation includes a variable, p, which represents an unknown number we need to find. The goal of solving an equation is to isolate the variable – to get p all by itself on one side of the equals sign.

Think of it like this: we want to know what single value of p will make the equation true. To find that, we need to simplify the equation step by step, making sure we keep that balance intact. Now, let's look closely at the left side of our equation: 3p - 7 + p. We have a few terms here: 3p, -7, and p. The terms 3p and p are like terms because they both contain the variable p. This is a crucial observation because the first step in solving this equation involves combining these like terms. Combining like terms is a fundamental algebraic operation that simplifies expressions and makes equations easier to work with. So, with a clear understanding of the equation's structure and the concept of like terms, we're perfectly set up to tackle the first step in solving it. Keep this understanding in your mind as we move forward, and you'll see how each step logically follows the previous one.

The Crucial First Step Combining Like Terms

Okay, so we've established that the key to simplifying our equation, 3p - 7 + p = 13, lies in combining those like terms. Remember, like terms are those that have the same variable raised to the same power. In our case, we have 3p and p. Think of 3p as "three lots of p" and p as "one lot of p." When we combine them, we're essentially adding those "lots" together. So, what happens when we add three p's and one p? We get four p's! Mathematically, this looks like 3p + p = 4p. This might seem simple, and that's because it is! But it's a vital step. By combining like terms, we're reducing the complexity of the equation and bringing it closer to a form that's easier to solve. Now, let's plug this back into our original equation. We had 3p - 7 + p = 13. After combining 3p and p, we get 4p - 7 = 13. See how much cleaner that looks? We've effectively reduced the number of terms on the left side, making the equation less cluttered and more manageable. This is a classic algebraic technique, and you'll use it all the time when solving equations. So, remember the power of combining like terms – it's your secret weapon for simplifying complex expressions. We're one step closer to the final solution, and this first step was a big one!

Analyzing the Answer Choices

Now that we've performed the first step combining like terms in the equation 3p - 7 + p = 13, which resulted in 4p - 7 = 13, let's carefully examine the answer choices provided. This is a critical part of the problem-solving process because it helps us ensure that we've not only performed the correct operation but also that we've interpreted the question accurately. The answer choices are:

• A. p - 7 = 13 - 3p
• B. 2p - 7 = 13
• C. 3p - 7 = 13 - p
• D. 4p - 7 = 13

Our goal is to identify which of these equations matches the result we obtained after combining like terms. Let's go through each option systematically. Option A, p - 7 = 13 - 3p, looks quite different from our result. It involves moving terms across the equals sign, which is a valid algebraic operation but not the first step we took. So, we can eliminate this option. Option B, 2p - 7 = 13, also doesn't match our result. The coefficient of p is incorrect; we arrived at 4p, not 2p. Therefore, we can rule out this choice as well. Option C, 3p - 7 = 13 - p, is similar to the original equation but with the p term moved to the right side. Again, this isn't the result of simply combining like terms on the left side. So, this option is also incorrect. Finally, we arrive at Option D, 4p - 7 = 13. This equation perfectly matches the result we obtained after combining 3p and p: 4p - 7 = 13. Therefore, Option D is the correct answer. By carefully comparing our result with each answer choice, we've not only confirmed our solution but also reinforced our understanding of the problem-solving process. This methodical approach is essential for tackling more challenging math problems in the future.

Why Option D is the Correct Answer

To really nail down why Option D is the only correct answer, let's recap our steps and think about what we've accomplished. We started with the equation 3p - 7 + p = 13. The first step, as we've emphasized, is to combine like terms. We identified 3p and p as the like terms on the left side of the equation. Adding these together, 3p + p, gives us 4p. This is the heart of the matter. We've simplified the left side of the equation by reducing two terms into one. Now, let's put this back into the original equation. We replace 3p + p with 4p, and the equation becomes 4p - 7 = 13. This is exactly what Option D states. No other answer choice reflects this specific step. Options A, B, and C involve different algebraic manipulations, such as moving terms across the equals sign or incorrectly combining the p terms. These are valid steps in solving equations eventually, but they aren't the first step in this particular problem.

Option D directly and accurately represents the result of combining like terms, which is the fundamental first step in simplifying this equation. This highlights the importance of understanding the order of operations and the specific instructions of the problem. We weren't asked to solve the equation completely; we were asked to identify the equation that results from the first step. By focusing on this specific instruction and applying the correct algebraic technique, we arrived confidently at the correct answer. Remember, in math, precision and attention to detail are key. Understanding why an answer is correct is just as important as knowing that it's correct.

Stepping Beyond the First Step What's Next?

So, we've successfully identified the resulting equation after the first step, which is 4p - 7 = 13. Awesome! But what if we wanted to solve for p completely? What would the next steps look like? This is a great way to extend our understanding and see how this initial simplification fits into the bigger picture of equation solving. Think of it as building on our foundation. We've laid the first brick, and now we're ready to add more. To isolate p, we need to get it all by itself on one side of the equation. Right now, it's attached to the -7. The golden rule of equation solving is that we can do the same thing to both sides of the equation without changing its balance. So, to get rid of the -7, we can add 7 to both sides. This gives us 4p - 7 + 7 = 13 + 7, which simplifies to 4p = 20. We're getting closer! Now, p is being multiplied by 4. To undo this multiplication, we divide both sides of the equation by 4. This gives us 4p / 4 = 20 / 4, which simplifies to p = 5. And there you have it! We've solved for p.

By taking these extra steps, we see how combining like terms is just the beginning. It's a crucial simplification that sets us up for the rest of the solution. This process of isolating the variable by performing inverse operations (adding to undo subtraction, dividing to undo multiplication) is the core of equation solving. So, while we were only asked for the first step in this problem, understanding the subsequent steps gives us a more complete picture and builds our confidence in tackling similar problems. Keep practicing these techniques, and you'll become a master equation solver in no time!

Practice Makes Perfect More Equations to Try

Alright guys, we've thoroughly dissected the equation 3p - 7 + p = 13, focusing on that critical first step of combining like terms. But like with any skill, practice is key to truly mastering equation solving. The more you work through different problems, the more comfortable and confident you'll become. Think of it like learning a new language the more you speak it, the more fluent you become. So, to help you on your journey to equation-solving mastery, let's look at a few more examples that use similar concepts. These examples will give you the chance to apply what you've learned and identify those initial steps for simplification. Remember, the goal isn't just to get the right answer but to understand the process behind it. Let's start with a simple variation:

• Example 1: Consider the equation 5x + 2 - 2x = 11. What is the resulting equation after the first step in the solution?

In this case, you'll need to identify the like terms involving x and combine them. This is very similar to what we did with the p terms in our original problem. Next up, let's try one with a little twist:

• Example 2: Consider the equation 2y - 8 + 5y + 3 = 10. What is the resulting equation after the first step in the solution?

Here, you have two sets of like terms: the y terms and the constant terms. Remember to combine each set separately. And finally, let's tackle one that looks a bit more complex:

• Example 3: Consider the equation 4a + 9 - a - 6 = 15. What is the resulting equation after the first step in the solution?

This equation requires careful attention to signs and the coefficients of the variable a. Work through these examples step by step, focusing on identifying and combining like terms as your first move. Don't rush the process! Take your time, write out each step clearly, and check your work. By tackling these practice problems, you'll not only solidify your understanding of combining like terms but also build the problem-solving skills you need to excel in algebra and beyond.

Key Takeaways and Common Pitfalls to Avoid

Before we wrap up our exploration of the equation 3p - 7 + p = 13 and the art of combining like terms, let's highlight some key takeaways and discuss common pitfalls to avoid. This is like adding the finishing touches to our understanding, ensuring that we're not just solving problems but also learning from them.

First, the most important takeaway is the concept of like terms. Remember, you can only combine terms that have the same variable raised to the same power. 3p and p are like terms, but 3p and 3p² are not. This is a crucial distinction! Second, always pay close attention to the signs (+ or -) in front of each term. The sign is part of the term, and it affects how you combine them. A common mistake is to forget to include the negative sign when combining terms like -7. Third, remember that combining like terms is just the first step in solving an equation. It simplifies the equation and sets you up for the subsequent steps of isolating the variable. Now, let's talk about some common pitfalls. One frequent error is trying to combine terms that aren't alike. Another is making arithmetic mistakes when adding or subtracting coefficients. A simple way to avoid this is to write out each step clearly and double-check your calculations. A third pitfall is forgetting the order of operations. Remember, you typically want to simplify each side of the equation as much as possible before you start moving terms across the equals sign.

By keeping these takeaways in mind and avoiding these common pitfalls, you'll be well-equipped to tackle a wide range of equation-solving problems. Think of these tips as your equation-solving toolkit – use them wisely, and you'll be amazed at what you can accomplish. We've covered a lot of ground, from understanding the initial equation to exploring subsequent steps and practicing with new examples. You're now well on your way to mastering this essential algebraic skill!

So, guys, we've really dug deep into the equation 3p - 7 + p = 13! We started by understanding what the equation means, then we pinpointed the crucial first step of combining like terms. We walked through the process step by step, saw why Option D (4p - 7 = 13) is the correct answer, and even looked ahead to the next steps in solving for p. We then tackled additional practice problems and highlighted key takeaways and pitfalls to avoid. That's a lot of equation-solving goodness! The key message here is that even complex-looking problems can be broken down into manageable steps. By mastering fundamental skills like combining like terms, you build a solid foundation for more advanced math.

Remember, math isn't about memorizing formulas; it's about understanding why things work. And when you understand the "why," you can confidently tackle any problem that comes your way. So, keep practicing, keep asking questions, and keep exploring the fascinating world of mathematics. You've got this!