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

Hey guys! Let's break down this math problem step-by-step. We're going to look at the equation 3p - 7 + p = 13 and figure out what the equation looks like after we take the very first step to solve it. Don't worry, it's easier than it might seem! We'll walk through the process together, making sure you understand each move. So, let's dive in and get started on solving this equation!

Understanding the Equation

Okay, so our starting point is the equation 3p - 7 + p = 13. The main goal when solving any equation like this is to isolate the variable, which in this case is p. That means we want to get p all by itself on one side of the equation. To do that, we need to simplify the equation by combining like terms. Like terms are those that have the same variable raised to the same power (in this case, terms with just p) or are constants (just numbers).

In our equation, we have two terms that involve p: 3p and +p. We also have a constant term, -7, on the left side of the equation and another constant term, 13, on the right side. Before we can start moving things around, we need to combine those p terms. This is a crucial first step because it simplifies the equation and makes it easier to work with. By combining like terms, we're essentially tidying up the equation and getting it ready for the next steps in solving for p. Remember, think of it as organizing your tools before starting a project – it makes the whole process smoother!

The First Step: Combining Like Terms

The golden rule here is to combine the terms that are alike. In our equation, 3p and +p are like buddies, so let's bring them together. Think of 3p as having three 'p's and +p as having one 'p'. If we add them, we get four 'p's, right? So, 3p + p equals 4p. That's the first key move! We've just simplified the left side of the equation by combining those p terms. This is a super common step in algebra, and mastering it will help you solve all sorts of equations. It's like learning the basics of a new language – once you understand the grammar, you can start forming sentences.

Now, let's rewrite the equation with our combined terms. Instead of 3p - 7 + p = 13, we now have 4p - 7 = 13. See how much cleaner that looks? We've taken the original equation and made it simpler by performing this single, but crucial, step. By combining like terms, we've essentially condensed the information, making the equation easier to grasp and manipulate. This is often the secret to making complex problems more manageable – break them down into smaller, simpler steps!

Analyzing the Resulting Equation

Alright, so after our first step of combining like terms, we've transformed the equation 3p - 7 + p = 13 into 4p - 7 = 13. This new equation is much easier to deal with! We've effectively reduced the number of terms on the left side, making it clearer what our next steps should be to isolate p. Now, let's really think about what this new equation, 4p - 7 = 13, means. It tells us that if we multiply some number p by 4 and then subtract 7, we'll end up with 13. Our goal now is to figure out what that number p is.

Looking at the equation 4p - 7 = 13, we can see that we're one step closer to getting p by itself. We still have that -7 hanging around on the left side, but we'll tackle that next. The key thing is to recognize that we've made progress. We've taken the original equation, which might have looked a bit daunting at first, and simplified it into a more manageable form. This is a great example of how breaking down a problem into smaller, logical steps can make even the trickiest equations seem less intimidating. By focusing on one step at a time, we're building a solid foundation for solving the entire problem. Remember, math is like building with blocks – each step builds upon the previous one!

Why This Step Matters

You might be wondering, "Why did we even bother combining like terms in the first place?" Well, this step is super important because it sets us up for success in the rest of the problem. By simplifying the equation, we've made it much easier to see what operations we need to perform to isolate p. Imagine trying to solve the original equation 3p - 7 + p = 13 without combining the p terms first. It would be like trying to bake a cake without measuring your ingredients – things could get messy and confusing fast!

Combining like terms is a fundamental skill in algebra, and it's something you'll use over and over again. It's not just about getting the right answer; it's about developing a clear and organized approach to problem-solving. When you simplify an equation, you're essentially creating a roadmap for yourself, showing you the most efficient way to reach the solution. This is why teachers and textbooks emphasize this step so much – it's the foundation upon which more complex algebraic concepts are built. So, mastering this skill is like unlocking a secret level in your math journey!

Identifying the Correct Option

Okay, let's bring it back to the original question. We were asked what the resulting equation is after the first step in the solution. We know that first step was combining the like terms 3p and +p to get 4p. This transformed our equation from 3p - 7 + p = 13 to 4p - 7 = 13. Now, let's think about the answer choices we might be given. We're looking for the one that matches this simplified equation. Sometimes, the answer choices might try to trick you by showing other steps in the solution or by rearranging the terms in a confusing way.

But we're smarter than that! We know that the correct answer will be the equation we got after combining like terms: 4p - 7 = 13. So, when you're faced with multiple-choice questions like this, always take the time to work through the steps yourself first. Don't just guess or try to spot the answer right away. By solving the problem step-by-step, you can be confident that you've arrived at the correct equation. Then, you can simply match your result to the answer choices and pick the winner. It's like being a detective, gathering clues and solving the mystery!

Next Steps in Solving for 'p'

So, we've nailed the first step and gotten our equation to 4p - 7 = 13. Awesome! But we're not done yet – we still need to actually solve for p. Think of this as the next level in our math game! The key to solving for p is to isolate it completely on one side of the equation. We've already combined like terms, which was a big step, but we still have that -7 hanging around. Our next move is to get rid of that -7. Remember, whatever we do to one side of the equation, we have to do to the other side to keep things balanced.

To get rid of the -7, we can add 7 to both sides of the equation. This is because -7 + 7 equals 0, so the -7 will disappear from the left side. If we add 7 to both sides, our equation becomes 4p - 7 + 7 = 13 + 7, which simplifies to 4p = 20. See how we're getting closer to isolating p? We've eliminated the constant term from the left side, leaving us with just 4p. This is like peeling away the layers of an onion – we're gradually stripping away the extra stuff until we get to the core, which is p.

Now, we have 4p = 20. This means that 4 times p equals 20. To get p by itself, we need to undo that multiplication. The opposite of multiplication is division, so we'll divide both sides of the equation by 4. This gives us 4p / 4 = 20 / 4, which simplifies to p = 5. Boom! We've solved for p! This whole process, from combining like terms to isolating the variable, is what solving an equation is all about. It's like following a recipe – each step is important, and if you follow them in the right order, you'll get the delicious result (in this case, the value of p).

Key Takeaways

Okay, guys, let's recap what we've learned! We started with the equation 3p - 7 + p = 13 and walked through the first crucial step in solving it: combining like terms. We identified that 3p and +p were like terms and combined them to get 4p, resulting in the equation 4p - 7 = 13. This step is super important because it simplifies the equation and makes it easier to work with. Remember, combining like terms is like tidying up your workspace before starting a project – it sets you up for success!

We also talked about why this step matters. It's not just about getting the right answer; it's about developing a clear and organized approach to problem-solving. Simplifying an equation is like creating a roadmap for yourself, showing you the most efficient way to reach the solution. And we even looked ahead to the next steps in solving for p, which involve isolating the variable by performing inverse operations (adding to get rid of subtraction, dividing to get rid of multiplication).

By understanding these concepts and practicing these steps, you'll become a master equation solver in no time! Math might seem intimidating at first, but when you break it down into smaller, manageable steps, it becomes much less scary and even kind of fun. So keep practicing, keep asking questions, and remember that every problem is just a puzzle waiting to be solved.