Solving For P In The Equation -3p + 1/8 = -1/4

In this article, we will delve into solving a fundamental algebraic equation. We aim to find the value of p that satisfies the equation -3p + 1/8 = -1/4. This is a common type of problem encountered in basic algebra, and mastering the technique to solve it is crucial for understanding more complex mathematical concepts. Understanding how to manipulate equations and isolate variables is a core skill in mathematics. This article provides a step-by-step guide for solving this particular equation, enhancing your understanding of algebraic manipulations. By the end of this guide, you'll be able to confidently solve similar equations and strengthen your algebraic foundation. We'll break down the problem into manageable steps, explaining the reasoning behind each operation, and ensuring that the solution is clear and easy to follow. So, let’s embark on this mathematical journey and unravel the value of p together!

Understanding the Equation

Before diving into the solution, let's first understand the components of the equation -3p + 1/8 = -1/4. This equation is a linear equation, where p is the variable we want to find. The equation states that negative three times p, plus one-eighth, equals negative one-quarter. This mathematical sentence sets the stage for our problem-solving endeavor. The left-hand side of the equation, -3p + 1/8, represents an expression involving the variable p, while the right-hand side, -1/4, is a constant value. Our task is to isolate p on one side of the equation to determine its value. This understanding is the cornerstone for solving any algebraic equation. Identifying the variable and understanding its relationship with other constants in the equation is crucial. The goal is to perform operations that maintain the equality of the equation while gradually isolating p. With this foundational understanding, we're ready to begin the process of solving for p.

Step 1: Isolate the Term with p

The first step in solving for p is to isolate the term containing p, which is -3p. To do this, we need to eliminate the constant term on the left side of the equation, which is 1/8. The concept of isolating the variable term is a fundamental algebraic technique. It involves strategically manipulating the equation to get the variable term by itself on one side. To eliminate 1/8, we subtract 1/8 from both sides of the equation. This is because performing the same operation on both sides maintains the equality. Subtracting 1/8 from both sides is a crucial step in simplifying the equation. This maintains the balance of the equation while moving us closer to isolating p. The equation now transforms to -3p = -1/4 - 1/8. This simplified form makes it easier to work with and prepares us for the next step in the solution process. The goal is to continue applying these operations until p stands alone on one side of the equation.

Step 2: Simplify the Right Side

Now, let's simplify the right side of the equation, which is -1/4 - 1/8. To subtract these fractions, we need a common denominator. The least common denominator for 4 and 8 is 8. Finding a common denominator is a crucial step in adding or subtracting fractions. It allows us to combine the fractions into a single term. We convert -1/4 to an equivalent fraction with a denominator of 8. To do this, we multiply both the numerator and the denominator of -1/4 by 2, resulting in -2/8. Now, the equation on the right side becomes -2/8 - 1/8. Subtracting the fractions, we get -3/8. Thus, the equation simplifies to -3p = -3/8. This simplification is a key step in isolating p. By combining the fractional terms, we've made the equation more manageable and closer to our final solution.

Step 3: Solve for p

We now have the equation -3p = -3/8. To solve for p, we need to isolate it by dividing both sides of the equation by -3. Dividing both sides of the equation by the coefficient of p is the final step in solving for the variable. This isolates p and reveals its value. Dividing -3p by -3 gives us p. On the right side, we have -3/8 divided by -3. Dividing a fraction by a number is the same as multiplying the fraction by the reciprocal of the number. The reciprocal of -3 is -1/3. So, we have (-3/8) * (-1/3). Multiplying these fractions, we get 3/24, which simplifies to 1/8. Therefore, p = 1/8. This completes the solution process. We have successfully isolated p and determined its value.

Final Answer

The value of p that makes the equation -3p + 1/8 = -1/4 true is 1/8. This is the solution to our problem. We have systematically worked through the equation, step by step, to arrive at this answer. We started by understanding the equation, then isolated the term with p, simplified the equation, and finally solved for p. This step-by-step approach is a valuable strategy for solving algebraic equations. It allows us to break down complex problems into manageable parts. By following these steps, you can confidently solve similar equations and strengthen your understanding of algebra. The solution p = 1/8 is the result of careful and methodical algebraic manipulation. Understanding the process is just as important as arriving at the correct answer.

Verification

To ensure our solution is correct, we can substitute p = 1/8 back into the original equation and verify that it holds true. Substituting p = 1/8 into the equation -3p + 1/8 = -1/4, we get -3(1/8) + 1/8 = -1/4. Multiplying -3 by 1/8, we get -3/8. So, the equation becomes -3/8 + 1/8 = -1/4. Adding the fractions on the left side, we get -2/8, which simplifies to -1/4. Thus, the equation -1/4 = -1/4 holds true. This verification step is crucial in ensuring the accuracy of our solution. It confirms that the value we found for p indeed satisfies the original equation. This practice of verification is a valuable habit in mathematics. It helps to catch errors and build confidence in your problem-solving abilities. By verifying our solution, we can be certain that p = 1/8 is the correct answer.

Conclusion

In conclusion, we have successfully solved the equation -3p + 1/8 = -1/4 and found that p = 1/8. This problem demonstrates the fundamental techniques of solving linear equations in algebra. We started by understanding the equation, then isolated the term with p, simplified the equation, and finally solved for p. This process highlights the importance of breaking down complex problems into manageable steps. By following a systematic approach, we can confidently tackle algebraic equations and arrive at accurate solutions. Furthermore, we verified our solution to ensure its correctness. This practice reinforces the importance of accuracy and attention to detail in mathematics. Mastering these fundamental skills will lay a strong foundation for further mathematical studies. The ability to solve equations like this is crucial for understanding more advanced concepts in algebra and beyond. Keep practicing and refining your problem-solving skills, and you'll be well-equipped to tackle any mathematical challenge that comes your way.