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

In the realm of mathematics, equations serve as fundamental tools for modeling and solving real-world problems. Understanding how to manipulate and solve equations is a crucial skill, and this article delves into the process of solving a specific equation step-by-step. We will dissect the equation 3p - 7 + p = 13, meticulously outlining each stage involved in isolating the variable p and arriving at the solution. This comprehensive guide will not only provide the answer but also illuminate the underlying principles of equation solving, empowering you to tackle similar mathematical challenges with confidence.

Problem Restatement: Simplifying the Equation

The initial challenge lies in deciphering the equation 3p - 7 + p = 13. Our primary objective is to determine the value of the variable p that satisfies this equation. To achieve this, we must embark on a journey of simplification and manipulation, carefully applying mathematical principles to isolate p on one side of the equation. The first step in this process involves identifying like terms and combining them to streamline the equation. Like terms are those that share the same variable raised to the same power. In this case, we have two terms containing the variable p: 3p and p. These terms can be combined to create a simpler expression. By combining like terms, we effectively reduce the complexity of the equation, paving the way for subsequent steps in the solution process. This initial simplification lays the groundwork for the remainder of the solution, making the equation more manageable and easier to solve. Remember, the goal is to isolate p, and combining like terms is a crucial step in that direction. The result of this step will be a more concise equation that brings us closer to the final answer.

Step 1: Combining Like Terms

In this section, we delve into the critical first step of simplifying the equation 3p - 7 + p = 13: combining like terms. As previously mentioned, like terms are those that share the same variable raised to the same power. In our equation, we have two terms containing the variable p: 3p and p. To combine these terms, we simply add their coefficients. The coefficient of 3p is 3, and the coefficient of p is understood to be 1 (since p is the same as 1p). Adding these coefficients, we get 3 + 1 = 4. Therefore, combining 3p and p results in the single term 4p. Now, let's rewrite the original equation with the combined like terms: 4p - 7 = 13. This new equation is significantly simpler than the original, making it easier to proceed with the solution. The essence of this step lies in reducing the number of terms in the equation, which in turn simplifies the subsequent steps. By combining like terms, we have effectively condensed the equation, making it more manageable and bringing us closer to isolating the variable p. This is a fundamental technique in equation solving, and mastering it is essential for tackling more complex mathematical problems.

Examining the Answer Choices

Having successfully executed the first step in solving the equation 3p - 7 + p = 13, we arrive at the simplified form 4p - 7 = 13. Now, let's shift our focus to the provided answer choices to identify the one that matches our result. Answer choice A, p - 7 = 13 - 3p, clearly does not match our simplified equation. This option involves rearranging terms and is not the result of the initial step of combining like terms. Answer choice B, 2p - 7 = 13, is also incorrect. While it does contain the term "- 7 = 13", the coefficient of p is 2, which does not align with our result of 4p. Answer choice C, 3p - 7 = 13 - p, is similarly incorrect. Like option A, this choice involves rearranging terms and does not represent the direct outcome of combining like terms in the original equation. Finally, answer choice D, 4p - 7 = 13, precisely matches our simplified equation. This option accurately reflects the result of combining the like terms 3p and p in the original equation. Therefore, after careful examination, we can confidently conclude that answer choice D is the correct representation of the equation after the first step in the solution.

Solution: The Resulting Equation

After meticulously combining the like terms in the given equation 3p - 7 + p = 13, we have arrived at the simplified equation 4p - 7 = 13. This transformation represents the crucial first step in isolating the variable p and solving for its value. By adding the coefficients of the p terms (3 and 1), we condensed the equation, making it more manageable for subsequent steps. This process of combining like terms is a fundamental technique in algebra, allowing us to simplify expressions and equations. Now, we can confidently identify the correct answer choice, which is D. 4p - 7 = 13. This equation represents the direct result of the initial simplification, setting the stage for further algebraic manipulations to determine the value of p. Understanding this first step is paramount, as it lays the foundation for the entire solution process. By mastering the art of combining like terms, you equip yourself with a powerful tool for tackling a wide range of algebraic equations.

Conclusion: Mastering Equation Solving

In conclusion, by carefully analyzing the equation 3p - 7 + p = 13 and executing the initial step of combining like terms, we successfully arrived at the resulting equation 4p - 7 = 13. This process underscores the importance of simplifying equations as a crucial first step in solving them. By combining like terms, we reduce the complexity of the equation, making it easier to manipulate and isolate the variable. This fundamental technique is a cornerstone of algebra and is essential for solving a wide range of mathematical problems. Moreover, this exercise highlights the significance of carefully examining answer choices and comparing them to our derived results. By systematically evaluating each option, we can confidently identify the correct solution. Remember, equation solving is a skill that improves with practice. By consistently applying the principles outlined in this article, you can strengthen your mathematical abilities and approach equation-solving challenges with greater confidence. The journey of mastering mathematics is paved with understanding fundamental concepts and applying them diligently, and this step-by-step solution serves as a valuable guide in that journey.

Therefore, the correct answer is D. 4p - 7 = 13.