Solving Equations: A Step-by-Step Guide

Oct 31, 2025 by Admin 40 views

Hey math enthusiasts! Let's dive into the fascinating world of equations. Specifically, we'll break down how to solve an equation and uncover the resulting equation after the first step in the solution to the equation: $3p - 7 + p = 13$ . Don't worry, it's not as scary as it sounds! We'll make it super clear, so you'll be acing these problems in no time. This guide is all about equations and how to solve them, so let's get started, shall we?

Understanding the Basics: Equations and Variables

Alright, before we jump into the nitty-gritty, let's make sure we're all on the same page. What even is an equation? Think of it like a balanced scale. An equation is a mathematical statement that shows two expressions are equal. It's like saying, "this side weighs the same as that side." The key element is the equal sign (=), which is the fulcrum of our scale. Everything on the left side of the equal sign must have the same value as everything on the right side. Got it? Awesome.

Now, what about that mysterious letter, 'p' in our equation? That's what we call a variable. It's a placeholder for a number we don't know yet. Our mission when solving an equation is to find the value of that variable that makes the equation true. So, we're basically playing a detective game, trying to uncover the hidden value of 'p'. Pretty cool, right? In our case, the equation we're tackling is $3p - 7 + p = 13$ . This means we have some 'p's, a number being subtracted, and another number on the other side of the equal sign. Our goal? Isolate 'p' and find its value. Seems like a fun adventure.

Here, we are dealing with basic algebraic equations, which involve combining like terms and isolating the variable. These skills are fundamental in mathematics and are used throughout various levels of study. So, understanding them now will benefit you in the long run. We're not dealing with complex equations or functions; rather, the focus is on mastering the fundamental steps to solve these kinds of equations.

Now, let's consider the equation $3p - 7 + p = 13$ to understand the first step.

The equation $3p - 7 + p = 13$

The First Step: Combining Like Terms

Okay, guys, let's get down to business! The very first step in solving the equation $3p - 7 + p = 13$ is to combine like terms. What does that mean? Well, like terms are terms that have the same variable raised to the same power. In our equation, we have two terms with the variable 'p': $3p$ and $+ p$ . We can combine them because they're both multiples of 'p'. Remember, 'p' is the same as $1p$ , so you can think of it as $3p + 1p$ . When we add these together, we get $4p$ . So, the first step is to simplify the left side of the equation by combining the 'p' terms. This will result in an equation that's easier to handle.

Why do we combine like terms? Because it simplifies the equation and makes it easier to work with. It's like organizing your desk before you start a project. Combining like terms reduces the number of terms we have to deal with, bringing us closer to isolating the variable and finding its value. The goal is always to simplify the equation to something that is easy to understand and solve. This initial step of simplification sets the stage for isolating the variable in subsequent steps. Combining like terms is a basic rule of algebra and an essential tool in equation solving.

Let's apply this in action: In our equation $3p - 7 + p = 13$ , the like terms are $3p$ and $p$ . Combining these gives us $4p$ . The rest of the equation remains the same for this step. That means we will have $4p - 7 = 13$ . See? Easy peasy.

Combining the like terms in the equation

The Resulting Equation After the First Step

So, what's the equation after the first step? As we've seen, combining the like terms $3p$ and $p$ in the equation $3p - 7 + p = 13$ simplifies the equation to $4p - 7 = 13$ . The correct answer here is D. $4p - 7 = 13$ . This is the equation we get after combining the 'p' terms. Remember, the goal of the first step is to simplify the equation, making it more manageable for further steps. That includes isolating the variable 'p'.

So, what does that mean for our equation? Originally, we had $3p - 7 + p = 13$ . After combining like terms, we get $4p - 7 = 13$ . This is now the starting point for our next step! We're one step closer to solving for 'p'. The process of solving an equation is like peeling away layers, where each step simplifies the problem until the variable is isolated. Knowing that the first step involves combining the like terms is a huge first move in mastering the equation solving.

Always remember to approach each step methodically and carefully. Doing this will build a strong foundation of your algebraic skills, making sure you will get the correct answer. Now we know, after the first step, our equation is much simpler. Keep this in mind when solving an equation. Every step plays a role in reaching the final solution. The first step, in most of the cases, will always be the simplification step.

The resulting equation after the first step

Why This Matters: Equations in the Real World

Okay, so we've solved this equation. But why does this all matter? Where does it fit into the world? Well, solving equations is a fundamental skill used in so many areas, like science, engineering, and even everyday life! Think about calculating the cost of groceries, figuring out the distance of a road trip, or even balancing a budget. Equations are there, helping us solve real-world problems.

In science and engineering, equations are used to model real-world phenomena, like the motion of objects, the flow of electricity, or the growth of populations. Equations allow scientists and engineers to predict and understand the world around us. In finance, equations help us with investing, calculating interest, or analyzing financial data. They are crucial for making informed decisions about money. And it's not just the big stuff! When you're following a recipe and need to adjust the quantities, you're using equations. Or, when you're planning a trip and calculating travel time, equations come into play. See? Equations are everywhere!

This simple skill of solving equations builds a strong foundation for more advanced mathematical concepts. So, by mastering the basics, you're equipping yourself with the tools to tackle more complex problems in the future. Knowing how to solve equations boosts your problem-solving skills, and helps you think logically and creatively. It's a skill that will serve you well in various fields, opening doors to new opportunities. So, keep practicing and exploring; the world of equations is full of amazing discoveries!

The real world implication of equations

Let's Recap: Key Takeaways

Alright, let's wrap things up with a quick recap. We started with the equation $3p - 7 + p = 13$ and wanted to know what happened after the first step. Here's what we learned:

Equations: Are mathematical statements that show two expressions are equal.
Variables: Are placeholders for unknown numbers (like our 'p').
Combining Like Terms: Is the process of adding or subtracting terms with the same variable raised to the same power.
The First Step: In our equation, the first step involved combining the like terms: $3p$ and $p$ . This simplifies the equation to $4p - 7 = 13$ .
Why It Matters: Solving equations is a fundamental skill, used in science, engineering, finance, and everyday life.

So there you have it! You've learned how to identify the first step in solving a simple algebraic equation. Keep practicing, and you'll become a pro at this in no time. The key is to remember the basics, take it step by step, and don't be afraid to ask for help when you need it. Math can be fun, and it can be rewarding. So go out there and conquer those equations! You've got this, guys!