Solving The Equation 2p - 4/3 - 1/2 = P - 4/6 - 32/2 A Step-by-Step Guide

Introduction

In the realm of mathematics, solving equations is a fundamental skill. The ability to manipulate equations, isolate variables, and arrive at solutions is crucial for various applications in science, engineering, and everyday problem-solving. This article delves into the step-by-step process of solving the equation 2p - 4/3 - 1/2 = p - 4/6 - 32/2 for the variable p. We will break down the equation, simplify terms, combine like terms, and ultimately determine the value of p that satisfies the equation. Understanding how to solve linear equations like this one is essential for mastering algebra and more advanced mathematical concepts. The equation presented involves fractions and requires careful handling of arithmetic operations to arrive at the correct solution. Let's embark on this mathematical journey and unravel the value of p.

Step 1: Simplify the Fractions

The initial step in solving the equation 2p - 4/3 - 1/2 = p - 4/6 - 32/2 involves simplifying the fractional terms. This simplification makes the equation easier to work with and reduces the chances of errors in subsequent steps. We begin by identifying the fractions present in the equation: 4/3, 1/2, 4/6, and 32/2. The fraction 32/2 can be immediately simplified to 16, as 32 divided by 2 equals 16. This simplification reduces the complexity of the equation. Next, we look at the fraction 4/6. This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2. Dividing 4 by 2 gives 2, and dividing 6 by 2 gives 3. Therefore, 4/6 simplifies to 2/3. Now our equation looks like this: 2p - 4/3 - 1/2 = p - 2/3 - 16. The remaining fractions, 4/3 and 1/2, are already in their simplest forms. Simplifying fractions is a critical step in solving equations as it makes the numbers smaller and easier to manage. This process not only aids in accurate calculations but also provides a clearer view of the equation's structure. By simplifying fractions, we pave the way for combining like terms and isolating the variable p.

Step 2: Find a Common Denominator

To effectively combine the fractional terms in the equation 2p - 4/3 - 1/2 = p - 2/3 - 16, we need to find a common denominator for the fractions. The fractions involved are 4/3, 1/2, and 2/3. To find the common denominator, we identify the least common multiple (LCM) of the denominators, which are 3 and 2. The multiples of 3 are 3, 6, 9, and so on, while the multiples of 2 are 2, 4, 6, 8, and so on. The least common multiple of 3 and 2 is 6. Therefore, 6 is the common denominator we will use to rewrite the fractions. Now, we convert each fraction to an equivalent fraction with a denominator of 6. For 4/3, we multiply both the numerator and the denominator by 2, resulting in (4 * 2) / (3 * 2) = 8/6. For 1/2, we multiply both the numerator and the denominator by 3, resulting in (1 * 3) / (2 * 3) = 3/6. For 2/3, we multiply both the numerator and the denominator by 2, resulting in (2 * 2) / (3 * 2) = 4/6. After converting the fractions, the equation becomes 2p - 8/6 - 3/6 = p - 4/6 - 16. Finding a common denominator is a crucial step because it allows us to add and subtract fractions easily. Without a common denominator, it is not possible to combine fractions directly. By converting all fractions to have the same denominator, we set the stage for simplifying the equation further and eventually solving for p. This step demonstrates the importance of understanding fraction manipulation in algebraic problem-solving.

Step 3: Combine Like Terms

After finding a common denominator and rewriting the fractions, the equation now reads 2p - 8/6 - 3/6 = p - 4/6 - 16. The next step is to combine the like terms on each side of the equation. On the left side, we have two fractional terms, -8/6 and -3/6, which can be combined. Adding these fractions, we get -8/6 - 3/6 = -11/6. So, the left side of the equation becomes 2p - 11/6. On the right side of the equation, we have -4/6 and -16. To combine these terms, we need to express -16 as a fraction with a denominator of 6. We can do this by multiplying -16 by 6/6, which gives us -96/6. Now, we can combine -4/6 and -96/6: -4/6 - 96/6 = -100/6. Therefore, the right side of the equation becomes p - 100/6. The equation now looks like this: 2p - 11/6 = p - 100/6. Combining like terms simplifies the equation by reducing the number of terms and making it easier to isolate the variable p. This step involves basic arithmetic operations with fractions, emphasizing the importance of mastering these skills in algebraic manipulations. By combining the fractional terms, we have streamlined the equation, bringing us closer to solving for p. This process demonstrates how simplifying expressions can lead to a more manageable equation, making the solution more accessible.

Step 4: Isolate the Variable p

Now that the equation has been simplified to 2p - 11/6 = p - 100/6, the next crucial step is to isolate the variable p. This involves rearranging the equation so that all terms containing p are on one side and all constant terms are on the other side. To begin, we can subtract p from both sides of the equation. This will move the p term from the right side to the left side. Subtracting p from both sides gives us: 2p - p - 11/6 = p - p - 100/6, which simplifies to p - 11/6 = -100/6. Next, we need to move the constant term -11/6 from the left side to the right side of the equation. To do this, we add 11/6 to both sides of the equation: p - 11/6 + 11/6 = -100/6 + 11/6. This simplifies to p = -100/6 + 11/6. Now, we can combine the fractions on the right side of the equation: -100/6 + 11/6 = -89/6. Therefore, the equation is now p = -89/6. Isolating the variable is a fundamental technique in solving equations. By performing the same operations on both sides of the equation, we maintain the equality while moving terms around to group like terms together. This process allows us to isolate the variable, making it possible to determine its value. In this case, we have successfully isolated p, leading us to the solution p = -89/6. This step highlights the importance of understanding inverse operations and their role in solving algebraic equations.

Step 5: Express the Solution

Having isolated the variable p, we have arrived at the solution p = -89/6. This solution can be expressed in different forms, depending on the context and the desired level of precision. The fraction -89/6 is an improper fraction, meaning the numerator is larger in magnitude than the denominator. We can convert this improper fraction to a mixed number to better understand its value. To convert -89/6 to a mixed number, we divide 89 by 6. The quotient is 14, and the remainder is 5. Therefore, -89/6 can be written as -14 and 5/6. So, p = -14 5/6. Alternatively, we can express the solution as a decimal. To convert -89/6 to a decimal, we divide -89 by 6. This gives us approximately -14.833. Therefore, p ≈ -14.833. Expressing the solution in different forms allows for flexibility in interpreting and applying the result. The improper fraction form, -89/6, is precise and useful for further calculations. The mixed number form, -14 5/6, provides a clearer sense of the magnitude of the solution. The decimal form, -14.833, offers a convenient way to compare the solution with other numerical values. In summary, the solution to the equation 2p - 4/3 - 1/2 = p - 4/6 - 32/2 is p = -89/6, which is equivalent to p = -14 5/6 and approximately equal to p = -14.833. This final step emphasizes the importance of being able to present solutions in various formats to suit different needs and preferences. Understanding how to convert between fractions, mixed numbers, and decimals is a valuable skill in mathematics and its applications.

Conclusion

In this comprehensive exploration, we have successfully solved the equation 2p - 4/3 - 1/2 = p - 4/6 - 32/2 for the variable p. The process involved a series of steps, each requiring careful attention to detail and a solid understanding of algebraic principles. We began by simplifying the fractions, which made the equation more manageable. Next, we found a common denominator to combine the fractional terms effectively. Then, we combined like terms on both sides of the equation to further simplify it. The crucial step of isolating the variable p followed, where we strategically moved terms around to get p by itself on one side of the equation. Finally, we expressed the solution in various forms, including an improper fraction, a mixed number, and a decimal, demonstrating the flexibility in representing mathematical results. The solution we arrived at is p = -89/6, which is equivalent to -14 5/6 and approximately -14.833. This exercise underscores the importance of mastering fundamental algebraic techniques, such as simplifying fractions, finding common denominators, combining like terms, and isolating variables. These skills are essential not only for solving equations but also for tackling more complex mathematical problems. Moreover, the ability to express solutions in different forms highlights the practical aspect of mathematics, where results need to be presented in a way that is clear, concise, and appropriate for the given context. By methodically working through each step, we have demonstrated how to approach and solve a linear equation involving fractions, providing a valuable framework for tackling similar problems in the future. The journey of solving this equation showcases the power of algebraic manipulation and the satisfaction of arriving at a solution through logical and systematic steps.