Solving 3p - 7 + P = 13 What Is The Resulting Equation After The First Step

In this article, we will delve into the process of solving the equation 3p - 7 + p = 13. Our primary goal is to identify the resulting equation after the initial step in the solution. This exercise is crucial for understanding the fundamental principles of algebraic manipulation and equation-solving techniques. We will meticulously break down each step, ensuring clarity and comprehension for readers of all backgrounds. Before diving into the solution, let's briefly discuss why solving equations is a vital skill in mathematics and its applications in various real-world scenarios. Mastering these techniques empowers us to tackle more complex problems and build a solid foundation in mathematical reasoning.

Why Equation Solving Matters

Equation solving is a cornerstone of mathematics, serving as a fundamental tool in numerous scientific, engineering, and everyday applications. From determining the trajectory of a rocket to calculating the optimal dosage of medication, equations are the language through which we model and understand the world around us. The ability to manipulate and solve equations is not just an academic exercise; it's a critical skill for problem-solving in a wide array of contexts. In this particular problem, we are presented with a linear equation, a common type encountered in various mathematical scenarios. Linear equations are characterized by a variable raised to the first power and can be solved using basic algebraic operations. Understanding how to solve them efficiently is essential for progressing to more advanced mathematical concepts. The process of solving equations involves isolating the variable of interest, which allows us to determine its value. This often requires applying a sequence of operations to both sides of the equation to maintain balance. In this article, we will focus on the first step of this process, emphasizing the importance of accurate and logical manipulations. By understanding the initial steps, we lay the groundwork for solving more complex equations and developing a deeper understanding of mathematical principles. Let’s proceed step-by-step.

Understanding the Given Equation

The equation we are tasked with solving is 3p - 7 + p = 13. This is a linear equation, meaning that the variable p is raised to the power of 1. Linear equations are among the most fundamental types of equations in algebra, and mastering their solution is crucial for progressing to more complex mathematical concepts. To solve this equation, our ultimate goal is to isolate the variable p on one side of the equation. This means we want to manipulate the equation until we have p equal to some value. The key to solving any equation is to maintain balance. Whatever operation we perform on one side of the equation, we must also perform on the other side. This ensures that the equality remains valid throughout the solution process. Before we begin any manipulations, it's essential to understand the structure of the equation. We have terms involving the variable p (3p and p), a constant term (-7), and a constant on the right side of the equation (13). Our first step will involve simplifying the equation by combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, 3p and p are like terms, and we can combine them to simplify the equation. This process of combining like terms is a fundamental step in solving equations and helps to reduce the complexity of the equation. By simplifying the equation, we make it easier to isolate the variable and find its value. Let's move on to the first step in the solution process.

Step 1: Combining Like Terms

The crucial first step in solving the equation 3p - 7 + p = 13 is to combine the like terms. In this equation, the like terms are 3p and p. Combining like terms involves adding or subtracting the coefficients of the terms that have the same variable raised to the same power. In this case, both 3p and p have the variable p raised to the power of 1, so we can combine them. To combine 3p and p, we add their coefficients. The coefficient of 3p is 3, and the coefficient of p is 1 (since p is the same as 1p). Adding these coefficients, we get 3 + 1 = 4. Therefore, combining 3p and p results in 4p. Now, we can rewrite the equation with the combined terms. The original equation was 3p - 7 + p = 13. After combining the like terms, the equation becomes 4p - 7 = 13. This simplified equation is the result of the first step in the solution process. By combining like terms, we have reduced the complexity of the equation and made it easier to isolate the variable p. This step is essential for progressing towards the final solution. It's important to note that we have maintained the balance of the equation by performing the same operation (combining like terms) on the left side only, as there were no like terms to combine on the right side. The next steps will involve further manipulations to isolate p, but for now, we have successfully completed the first step. The importance of this step cannot be overstated, as it lays the foundation for the subsequent steps and brings us closer to the solution.

Identifying the Resulting Equation

After performing the first step, which involves combining like terms, the resulting equation is 4p - 7 = 13. This equation is a simplified version of the original equation, making it easier to proceed with the solution. Let's compare this result with the given options to determine the correct answer. The options provided are:

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

By comparing our resulting equation, 4p - 7 = 13, with the options, it is clear that option D matches our result. Options A, B, and C do not match the equation we obtained after combining like terms. Option A involves rearranging terms in a way that is not consistent with the first step of solving the equation. Option B has an incorrect coefficient for the p term. Option C also involves rearranging terms incorrectly. Therefore, the correct answer is D. 4p - 7 = 13. This exercise highlights the importance of performing each step in the equation-solving process accurately. A single error in any step can lead to an incorrect result. By carefully combining like terms, we have successfully simplified the equation and identified the resulting equation after the first step. This demonstrates a fundamental skill in algebra and problem-solving. Understanding these basic steps is crucial for tackling more complex equations and mathematical problems in the future. In the next sections, we may explore the subsequent steps required to fully solve the equation and find the value of p, but for now, we have successfully identified the equation resulting from the initial step.

The Next Steps in Solving the Equation

While we have successfully identified the resulting equation after the first step, 4p - 7 = 13, it is beneficial to briefly discuss the subsequent steps required to fully solve for p. This will provide a more comprehensive understanding of the equation-solving process. After combining like terms, our equation is 4p - 7 = 13. The next goal is to isolate the term with the variable, which in this case is 4p. To do this, we need to eliminate the constant term, which is -7, from the left side of the equation. We can accomplish this by adding 7 to both sides of the equation. This maintains the balance of the equation and moves us closer to isolating p. Adding 7 to both sides of the equation gives us: 4p - 7 + 7 = 13 + 7. Simplifying this, we get 4p = 20. Now, we have the term with the variable isolated on one side of the equation. The final step is to isolate p itself. Since p is multiplied by 4, we need to perform the inverse operation, which is division. We divide both sides of the equation by 4 to solve for p. Dividing both sides by 4, we get: 4p / 4 = 20 / 4. Simplifying this, we find p = 5. Therefore, the solution to the equation 3p - 7 + p = 13 is p = 5. This brief overview of the subsequent steps demonstrates the complete process of solving the equation. By combining like terms, adding the constant to both sides, and dividing by the coefficient, we successfully isolated the variable and found its value. Understanding these steps is crucial for solving various algebraic equations and building a strong foundation in mathematics. While the primary focus of this article was on the first step, it is valuable to see how the subsequent steps build upon this foundation to arrive at the final solution.

Conclusion

In conclusion, the first step in solving the equation 3p - 7 + p = 13 involves combining like terms, which results in the equation 4p - 7 = 13. This step is crucial for simplifying the equation and paving the way for isolating the variable p. We meticulously walked through the process, emphasizing the importance of accuracy and balance in equation manipulation. The correct answer, as identified from the given options, is D. 4p - 7 = 13. Furthermore, we briefly discussed the subsequent steps required to fully solve the equation, demonstrating how the solution process unfolds step-by-step. This comprehensive approach provides a solid understanding of equation-solving techniques and reinforces the importance of each step in the process. Mastering these fundamental skills is essential for success in mathematics and its applications in various fields. By understanding the underlying principles and practicing diligently, one can develop confidence and proficiency in solving equations. The ability to solve equations is not only a valuable mathematical skill but also a critical tool for problem-solving in everyday life and various professional domains. Therefore, a strong foundation in equation solving is a worthwhile investment in one's mathematical journey.