Solving For Q In 3(q + P) = 5 + 7q A Step-by-Step Guide

In the realm of algebra, rearranging formulas to isolate a specific variable is a fundamental skill. This process, known as making a variable the subject of the formula, allows us to express one variable in terms of others, providing a powerful tool for solving equations and understanding relationships between variables. In this comprehensive guide, we will delve into the step-by-step process of making q the subject of the formula 3(q + p) = 5 + 7q, ultimately expressing the solution in the form (ap - b)/c, where a, b, and c are positive integers. This exercise not only reinforces algebraic manipulation techniques but also highlights the importance of precision and clarity in mathematical problem-solving. This guide will provide a detailed, step-by-step solution, ensuring clarity and understanding at each stage. We will break down the algebraic manipulations involved, emphasizing the underlying principles and techniques. By the end of this guide, you will not only be able to solve this specific problem but also gain a deeper understanding of how to rearrange formulas effectively. This skill is crucial for various applications in mathematics, science, and engineering, where manipulating equations is a common task. So, let's embark on this algebraic journey and master the art of making q the subject of the formula.

1. Expanding the Parentheses: Laying the Foundation

The initial step in isolating q involves expanding the parentheses on the left side of the equation. This is achieved by applying the distributive property, which states that a(b + c) = ab + ac. Applying this property to our equation, 3(q + p) = 5 + 7q, we multiply 3 by both q and p, resulting in: 3q + 3p = 5 + 7q. Expanding the parentheses is a crucial first step because it removes the grouping, allowing us to work with individual terms and rearrange them more easily. Without this step, the q term inside the parentheses would be difficult to isolate. This expansion sets the stage for the subsequent steps, where we will gather the q terms on one side of the equation and the constant terms on the other. The distributive property is a cornerstone of algebraic manipulation, and mastering its application is essential for solving a wide range of equations. By expanding the parentheses correctly, we ensure that the equation remains balanced and that we are on the right track to isolating q. This step highlights the importance of understanding and applying fundamental algebraic principles to simplify complex expressions. Correctly applying the distributive property not only simplifies the equation but also lays a solid foundation for the subsequent steps in solving for q. This careful attention to detail is a hallmark of effective algebraic problem-solving. This expansion not only simplifies the equation but also makes it more manageable for further manipulation. It's a fundamental technique in algebra that unlocks the path to isolating the desired variable.

2. Gathering the q Terms: Unifying the Variable

Our next objective is to consolidate all terms containing q on one side of the equation. To achieve this, we subtract 3q from both sides of the equation 3q + 3p = 5 + 7q. This operation maintains the equality of the equation while effectively moving the 3q term from the left side to the right side. The result is: 3p = 5 + 4q. Subtracting 3q from both sides is a strategic move, as it allows us to group the q terms together, simplifying the equation and bringing us closer to isolating q. This step demonstrates the importance of performing the same operation on both sides of an equation to maintain balance. It's a fundamental principle of algebraic manipulation that ensures the equation remains valid throughout the solving process. By gathering the q terms on one side, we reduce the complexity of the equation and create a clearer path towards isolating q. This step highlights the power of strategic algebraic manipulation in simplifying equations and making them easier to solve. The key idea here is to strategically manipulate the equation to bring like terms together, which is a crucial technique in solving algebraic equations. By subtracting 3q from both sides, we've taken a significant step towards isolating q and making it the subject of the formula. This process not only simplifies the equation but also makes it more visually clear and easier to work with.

3. Isolating the q Term: Preparing for the Final Step

Now, we aim to isolate the term containing q on one side of the equation. In the equation 3p = 5 + 4q, the q term is currently accompanied by the constant term 5. To isolate 4q, we subtract 5 from both sides of the equation. This operation maintains the balance of the equation while effectively removing the constant term from the right side. The resulting equation is: 3p - 5 = 4q. Subtracting 5 from both sides is a crucial step in isolating the q term, as it eliminates the constant term that is hindering our progress. This step further simplifies the equation and brings us closer to our goal of making q the subject of the formula. This process underscores the importance of performing inverse operations to isolate variables in algebraic equations. By subtracting 5, we are essentially undoing the addition of 5, thereby isolating the term containing q. This step demonstrates the power of strategic algebraic manipulation in simplifying equations and making them easier to solve. The principle of performing the same operation on both sides of the equation is paramount here, ensuring that the equation remains balanced and valid throughout the manipulation. Isolating the 4q term is a pivotal moment in the solution process. It sets the stage for the final step, where we will divide to completely isolate q and express it as the subject of the formula. This step showcases the strategic nature of algebraic manipulation, where each operation is carefully chosen to move closer to the desired solution.

4. Making q the Subject: The Final Transformation

The final step in making q the subject of the formula involves isolating q completely. In the equation 3p - 5 = 4q, q is currently being multiplied by 4. To isolate q, we divide both sides of the equation by 4. This operation maintains the balance of the equation while effectively undoing the multiplication by 4. The resulting equation is: q = (3p - 5) / 4. Dividing both sides by 4 is the definitive step in isolating q, as it completely removes the coefficient that was attached to q. This step represents the culmination of our algebraic manipulations, where we have successfully rearranged the formula to express q in terms of p. This process highlights the importance of understanding inverse operations in solving algebraic equations. By dividing, we are essentially undoing the multiplication, thereby isolating the variable of interest. This step demonstrates the power of strategic algebraic manipulation in simplifying equations and making them easier to interpret. The final form of the equation, q = (3p - 5) / 4, clearly expresses q as the subject of the formula, fulfilling the objective of the problem. This result can be used to calculate the value of q for any given value of p, showcasing the practical utility of making a variable the subject of a formula. The act of dividing both sides by 4 is a precise and deliberate algebraic move. It's the final flourish in our solution, transforming the equation into its desired form, where q stands alone, proudly declared as the subject of the formula. This step is the algebraic equivalent of a mic drop, signifying the successful completion of our task.

5. Expressing the Answer in the Required Form: Meeting the Criteria

Our solution, q = (3p - 5) / 4, is now in the form (ap - b)/c, where a, b, and c are positive integers. By comparing our solution to the required form, we can identify the values of a, b, and c. In this case, a = 3, b = 5, and c = 4. This final step ensures that our answer meets the specific criteria outlined in the problem statement. It demonstrates the importance of paying close attention to the instructions and presenting the solution in the requested format. This process highlights the need for precision and attention to detail in mathematical problem-solving. By explicitly identifying the values of a, b, and c, we provide a clear and unambiguous answer that satisfies the requirements of the problem. This step reinforces the importance of clear communication in mathematics, where presenting solutions in a standardized format enhances understanding and avoids ambiguity. The act of aligning our solution with the required form is a crucial step in the problem-solving process. It's the final polish, ensuring that our answer is not only correct but also presented in a way that is easily understood and accepted. This step showcases the importance of adhering to conventions and expectations in mathematical communication. Expressing the answer in the correct form is not merely a formality; it's an essential part of the problem-solving process. It demonstrates a thorough understanding of the question and the ability to present the solution in a clear, concise, and mathematically sound manner. This final step is the culmination of our efforts, providing a complete and satisfying solution to the problem.

Conclusion: Mastering Algebraic Manipulation

In conclusion, we have successfully made q the subject of the formula 3(q + p) = 5 + 7q and expressed the solution in the form (ap - b)/c, where a, b, and c are positive integers. This process involved a series of algebraic manipulations, including expanding parentheses, gathering like terms, isolating the q term, and dividing to make q the subject. Each step was carefully executed to maintain the balance of the equation and ensure the accuracy of the solution. This exercise demonstrates the importance of mastering algebraic manipulation techniques, which are fundamental to solving a wide range of mathematical problems. By understanding and applying these techniques, we can effectively rearrange formulas, solve equations, and gain deeper insights into the relationships between variables. This guide has provided a detailed, step-by-step solution, emphasizing the underlying principles and techniques involved. By following these steps, you can not only solve this specific problem but also develop the skills and confidence to tackle similar algebraic challenges. The ability to make a variable the subject of a formula is a valuable skill in mathematics, science, and engineering, where manipulating equations is a common task. By mastering this skill, you can enhance your problem-solving abilities and achieve greater success in your academic and professional pursuits. This journey through algebraic manipulation has highlighted the power and elegance of mathematical problem-solving. It's a testament to the logical and systematic approach that mathematics provides, allowing us to transform complex problems into manageable steps and arrive at clear and concise solutions. The successful completion of this task is not just about finding the answer; it's about the journey of understanding and applying mathematical principles to solve problems effectively.