Solving Equations With Fractions A Step-by-Step Guide To Solve (3/2)p - 7 + (5/4)p = 5 + (7/4)p
Introduction
In this comprehensive guide, we will delve into the step-by-step process of solving the equation (3/2)p - 7 + (5/4)p = 5 + (7/4)p for the variable p. This equation involves fractions and requires careful manipulation to isolate p and find its value. Mastering the techniques used here will be invaluable for tackling various algebraic problems. Whether you are a student learning algebra or someone looking to refresh your math skills, this detailed explanation will provide you with a clear understanding of the solution process. We will break down each step, explaining the reasoning behind it and highlighting key concepts along the way. By the end of this guide, you will be equipped with the knowledge and confidence to solve similar equations independently. Let's embark on this mathematical journey together and unlock the value of p!
Understanding the Equation
Before we begin solving the equation, let's take a moment to understand its structure. The equation (3/2)p - 7 + (5/4)p = 5 + (7/4)p is a linear equation in one variable, p. It involves fractions, which might seem daunting at first, but with a systematic approach, we can easily handle them. The goal is to isolate p on one side of the equation, which means we need to collect all terms containing p on one side and all constant terms on the other side. To achieve this, we will use several algebraic properties, such as the commutative, associative, and distributive properties, as well as the addition and multiplication properties of equality. Each of these properties allows us to manipulate the equation while maintaining its balance. The left-hand side of the equation contains terms with p and a constant, while the right-hand side also contains a term with p and a constant. Our strategy will be to combine like terms, eliminate fractions, and then isolate p. This process involves a series of steps, each building upon the previous one, to gradually simplify the equation until we arrive at the solution. Remember, the key to solving algebraic equations is to perform the same operations on both sides, ensuring that the equation remains balanced throughout the solution process. Understanding the structure of the equation and the properties we can use to manipulate it is crucial for success.
Step-by-Step Solution
Now, let's dive into the step-by-step solution of the equation (3/2)p - 7 + (5/4)p = 5 + (7/4)p. We will break down each step in detail, explaining the reasoning behind it. This methodical approach will help you understand not just the solution, but also the underlying principles of solving linear equations with fractions.
Step 1: Combine Like Terms on Each Side
The first step in simplifying the equation is to combine the terms with p on each side. On the left-hand side, we have (3/2)p and (5/4)p. To combine these, we need a common denominator, which in this case is 4. We can rewrite (3/2)p as (6/4)p. So, the left-hand side becomes (6/4)p + (5/4)p - 7. Combining the terms with p, we get (11/4)p - 7. The left side of the equation is now simplified to (11/4)p - 7. On the right-hand side, we only have one term with p, which is (7/4)p, and a constant term, 5. Thus, the right-hand side remains as 5 + (7/4)p. This step of combining like terms is crucial as it simplifies the equation and makes it easier to work with. By adding or subtracting coefficients of the same variable, we reduce the number of terms, which in turn reduces the complexity of the equation. Remember, combining like terms is a fundamental algebraic technique that is used extensively in solving various types of equations. Make sure you understand this step thoroughly before moving on to the next.
Step 2: Eliminate Fractions
Fractions can sometimes make equations look more complicated than they are. Therefore, the next step is to eliminate the fractions from the equation. To do this, we find the least common multiple (LCM) of the denominators, which in this case is 4. We then multiply every term in the equation by 4. This step is based on the multiplication property of equality, which states that if we multiply both sides of an equation by the same non-zero number, the equation remains balanced. Multiplying the entire equation (11/4)p - 7 = 5 + (7/4)p by 4, we get: 4 * [(11/4)p - 7] = 4 * [5 + (7/4)p]. Distributing the 4 on both sides, we have: (4 * (11/4)p) - (4 * 7) = (4 * 5) + (4 * (7/4)p). Simplifying each term, we get: 11p - 28 = 20 + 7p. Now, the equation is free of fractions, making it much easier to handle. Eliminating fractions is a powerful technique that simplifies the equation and allows us to focus on isolating the variable. This step is essential in solving equations that involve fractions and is a key skill in algebra. By multiplying every term by the LCM of the denominators, we ensure that the resulting equation is equivalent to the original one but without the fractions.
Step 3: Move Terms with p to One Side
Now that we have eliminated the fractions, we need to gather all terms containing p on one side of the equation. It's a matter of choice whether you move the p terms to the left or the right side. In this case, let's move the p terms to the left side. We can do this by subtracting 7p from both sides of the equation 11p - 28 = 20 + 7p. This step is based on the subtraction property of equality, which states that if we subtract the same quantity from both sides of an equation, the equation remains balanced. Subtracting 7p from both sides, we get: 11p - 28 - 7p = 20 + 7p - 7p. Simplifying, we have: 4p - 28 = 20. By subtracting 7p from both sides, we have successfully moved all terms with p to the left side of the equation. This step is crucial as it brings us closer to isolating p. Moving all terms with the variable to one side and constant terms to the other is a standard technique in solving algebraic equations. It allows us to consolidate the variable terms and prepare for the final steps of isolating the variable.
Step 4: Move Constant Terms to the Other Side
The next step is to move all the constant terms to the other side of the equation. We have the equation 4p - 28 = 20. To move the constant term -28 to the right side, we add 28 to both sides of the equation. This is based on the addition property of equality, which states that if we add the same quantity to both sides of an equation, the equation remains balanced. Adding 28 to both sides, we get: 4p - 28 + 28 = 20 + 28. Simplifying, we have: 4p = 48. Now, all the terms with p are on the left side, and all the constant terms are on the right side. This step is a significant milestone in isolating p. By strategically adding or subtracting constants, we can isolate the variable term on one side of the equation. This step, combined with the previous step of moving variable terms, sets the stage for the final step of solving for the variable.
Step 5: Isolate p
Finally, we need to isolate p to find its value. We have the equation 4p = 48. To isolate p, we divide both sides of the equation by the coefficient of p, which is 4. This is based on the division property of equality, which states that if we divide both sides of an equation by the same non-zero number, the equation remains balanced. Dividing both sides by 4, we get: (4p) / 4 = 48 / 4. Simplifying, we have: p = 12. Thus, the solution to the equation is p = 12. This final step completes the process of solving the equation. By dividing both sides by the coefficient of the variable, we successfully isolate the variable and find its value. This step is the culmination of all the previous steps, and it provides the answer to the original problem. Now, we have successfully found the value of p that satisfies the equation.
Verification
To ensure our solution is correct, it's always a good practice to verify it by substituting the value of p back into the original equation. Our solution is p = 12, and the original equation is (3/2)p - 7 + (5/4)p = 5 + (7/4)p. Substituting p = 12 into the equation, we get: (3/2)(12) - 7 + (5/4)(12) = 5 + (7/4)(12). Simplifying each term: (3 * 12) / 2 - 7 + (5 * 12) / 4 = 5 + (7 * 12) / 4. 36 / 2 - 7 + 60 / 4 = 5 + 84 / 4. 18 - 7 + 15 = 5 + 21. 11 + 15 = 26. 26 = 26. Since the left-hand side equals the right-hand side, our solution p = 12 is correct. Verification is an essential step in the problem-solving process. It confirms that the solution we obtained satisfies the original equation, thereby ensuring the correctness of our answer. By substituting the value back into the original equation, we can catch any errors that might have occurred during the solution process. This step provides confidence in the solution and reinforces the understanding of the equation.
Conclusion
In conclusion, we have successfully solved the equation (3/2)p - 7 + (5/4)p = 5 + (7/4)p for p. By following a step-by-step approach, we combined like terms, eliminated fractions, moved terms with p to one side, moved constant terms to the other side, and finally isolated p. The solution we found is p = 12. We also verified our solution by substituting it back into the original equation, confirming its correctness. This exercise demonstrates the importance of a systematic approach in solving algebraic equations. Each step is crucial and builds upon the previous one, leading us to the solution. Mastering these techniques will enable you to confidently tackle similar equations and enhance your problem-solving skills in algebra. Remember, the key to success in mathematics is practice and a clear understanding of the underlying principles. We hope this comprehensive guide has been helpful in your learning journey. Keep practicing and exploring new mathematical challenges!
