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

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#h1 Understanding the Equation 3p - 7 + p = 13

In this article, we will delve into the process of solving the algebraic equation 3p - 7 + p = 13. This equation is a linear equation in one variable, 'p', and our goal is to isolate 'p' on one side of the equation to find its value. To do this effectively, we'll break down the solution step by step, focusing on the initial move that simplifies the equation. Mastering the initial steps in solving algebraic equations is crucial for building a strong foundation in algebra.

Before diving into the solution, it's important to understand the basic principles of equation solving. The fundamental idea is to maintain the balance of the equation. Any operation performed on one side of the equation must also be performed on the other side to keep the equation true. This principle allows us to manipulate the equation while preserving its integrity. In this specific equation, we notice terms with the variable 'p' and constant terms. The initial step typically involves combining like terms or isolating the variable term. Let's explore the equation in detail and identify the most efficient first step to simplify it and move closer to the solution.

Combining Like Terms: The First Crucial Step

To solve the equation 3p - 7 + p = 13, the first step involves combining like terms. Like terms are terms that contain the same variable raised to the same power. In this equation, we have two terms with the variable 'p': 3p and +p. Combining these terms is a fundamental algebraic operation that simplifies the equation and makes it easier to solve. When combining like terms, we simply add or subtract their coefficients (the numbers in front of the variable). In our case, we have 3p + p, which can be seen as 3p + 1p. Adding the coefficients, 3 + 1, gives us 4. Therefore, 3p + p simplifies to 4p. This combination is a direct application of the distributive property in reverse, where we factor out the common variable 'p'. By combining like terms, we reduce the number of terms in the equation, which is a key strategy in simplifying algebraic expressions. This step is not just about making the equation look simpler; it's about making it mathematically more manageable. After combining like terms, the equation transforms from 3p - 7 + p = 13 to 4p - 7 = 13, which is a more concise and easier-to-handle form. The result of this initial step significantly streamlines the subsequent steps in solving for 'p'.

#h2 Detailed Explanation of Combining 'p' Terms

To fully grasp the concept of combining like terms, let's delve deeper into the mechanics of adding 3p and p. The term '3p' represents three times the variable 'p', while 'p' (or +p) represents one times the variable 'p'. When we add these together, we are essentially adding three 'p's to one 'p', resulting in a total of four 'p's, which we write as 4p. This is analogous to adding three apples to one apple, resulting in four apples. The variable 'p' acts as a placeholder for any number, and the coefficients tell us how many of these placeholders we have. The addition of like terms is a direct application of the distributive property, which states that a(b + c) = ab + ac. In reverse, we can think of 3p + 1p as p(3 + 1), which simplifies to p(4), or 4p. This understanding is crucial because it forms the basis for simplifying more complex algebraic expressions. The ability to quickly and accurately combine like terms is essential for solving equations efficiently. It reduces the complexity of the equation, making subsequent operations, such as isolating the variable, much easier. This foundational skill is not only applicable to linear equations but also extends to polynomials and other algebraic structures. By mastering the concept of combining like terms, students can approach algebraic problems with greater confidence and clarity.

#h2 The Transformed Equation: 4p - 7 = 13

After combining the like terms 3p and p, the original equation 3p - 7 + p = 13 transforms into a more simplified form: 4p - 7 = 13. This transformation is a crucial step in the solution process because it consolidates the variable terms into a single term, 4p. The equation now presents a clearer picture of the relationship between the variable 'p' and the constants. The simplified equation 4p - 7 = 13 is significantly easier to work with compared to the original equation. It has fewer terms and directly shows the variable term (4p) and the constant term (-7) on one side, with the constant 13 on the other side. This form allows us to proceed more directly towards isolating 'p'. The next logical step in solving this equation would be to isolate the term with 'p', which involves dealing with the constant term -7. To do this, we would add 7 to both sides of the equation, maintaining the balance and moving closer to isolating 'p'. This transformation highlights the importance of simplifying equations as a strategic move in problem-solving. A simplified equation is not only easier to look at, but it also reduces the chances of making errors in subsequent steps. The transformed equation 4p - 7 = 13 sets the stage for the final steps in solving for the value of 'p'.

#h2 Why the Other Options Are Incorrect

Understanding why the correct answer is 4p - 7 = 13 also involves recognizing why the other options are incorrect. Let's examine each of the incorrect options:

  • Option A: p - 7 = 13 - 3p

    This option suggests moving the 3p term to the right side of the equation by subtracting it. While it's a valid algebraic manipulation, it's not the first step in simplifying the given equation. The most efficient initial step is to combine like terms on the same side of the equation. Moving terms across the equals sign should typically follow the combination of like terms. This approach, while leading to the correct solution eventually, adds an unnecessary step at the beginning.

  • Option B: 2p - 7 = 13

    This option incorrectly combines the 'p' terms. It seems to subtract 'p' from 3p, resulting in 2p, which is a mathematical error. The correct operation is to add the coefficients of the 'p' terms (3p + 1p), which results in 4p, not 2p. This error demonstrates a misunderstanding of how to combine like terms, which is a fundamental concept in algebra. Miscalculations like these can lead to incorrect solutions, emphasizing the importance of accurately executing each step.

  • Option C: 3p - 7 = 13 - p

    Similar to Option A, this option involves moving the 'p' term to the right side of the equation. Again, while this is a valid algebraic operation, it's not the most efficient first step. The initial focus should be on simplifying the left side of the equation by combining the like terms 3p and +p. Moving terms across the equals sign prematurely complicates the equation rather than simplifying it. This approach deviates from the standard strategy of simplifying each side of the equation before manipulating terms across the equals sign.

Each of these incorrect options demonstrates a deviation from the optimal first step in solving the equation. The most efficient and mathematically sound first step is to combine the like terms 3p and p, resulting in 4p. Understanding why these options are incorrect reinforces the importance of following the correct order of operations and applying algebraic principles accurately.

#h3 Continuing the Solution: Isolating 'p'

Having established the correct first step and transformed the equation to 4p - 7 = 13, let's briefly outline the subsequent steps to completely solve for 'p'. The goal is to isolate 'p' on one side of the equation. To do this, we need to undo the operations that are being applied to 'p'. Currently, 'p' is being multiplied by 4, and 7 is being subtracted from the result.

The next step is to address the subtraction of 7. To undo this, we add 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 4p - 7 = 13 gives us:

4p - 7 + 7 = 13 + 7

This simplifies to:

4p = 20

Now, 'p' is being multiplied by 4. To undo this multiplication, we divide both sides of the equation by 4:

4p / 4 = 20 / 4

This simplifies to:

p = 5

Therefore, the solution to the equation 3p - 7 + p = 13 is p = 5. This complete solution demonstrates the step-by-step process of solving a linear equation, starting with the crucial first step of combining like terms and then systematically isolating the variable. This systematic approach is applicable to a wide range of algebraic equations and is a fundamental skill in mathematics. The ability to accurately and efficiently solve equations is essential for success in algebra and beyond.

#h3 Conclusion

In conclusion, the first and most efficient step in solving the equation 3p - 7 + p = 13 is to combine the like terms 3p and p, which results in the equation 4p - 7 = 13. This step simplifies the equation and sets the stage for isolating the variable 'p'. Understanding the importance of combining like terms as an initial step is crucial for efficiently solving algebraic equations. The incorrect options highlight common errors in algebraic manipulation, such as prematurely moving terms across the equals sign or incorrectly combining like terms. By mastering the correct order of operations and applying algebraic principles accurately, students can confidently solve a variety of equations. The subsequent steps involve isolating 'p' by adding 7 to both sides and then dividing by 4, leading to the solution p = 5. This step-by-step approach provides a clear and concise method for solving linear equations, a fundamental skill in mathematics and essential for more advanced algebraic concepts. Consistent practice and a thorough understanding of these principles will build a strong foundation in algebra.