Solving For Y In The Equation Q(a+y) = 67y + 93

Introduction

In this article, we delve into the realm of algebraic equations, specifically focusing on solving for the variable y in the equation q(a+y) = 67y + 93. This equation, seemingly simple at first glance, presents an opportunity to explore fundamental algebraic principles and techniques. Understanding how to isolate y and express it in terms of other variables (q and a in this case) is a crucial skill in mathematics and various scientific disciplines. This comprehensive guide aims to provide a step-by-step approach to solving this equation, ensuring clarity and a thorough understanding of the underlying concepts. Whether you're a student grappling with algebra or simply seeking to refresh your mathematical prowess, this article will serve as a valuable resource. We will break down the process into manageable steps, covering the distribution of terms, rearrangement of the equation, and the final isolation of y. So, let's embark on this algebraic journey and unravel the solution to this intriguing equation.

Breaking Down the Equation: Step-by-Step Solution

Our journey to solve for y in the equation q(a+y) = 67y + 93 begins with a meticulous breakdown of each step. The initial equation presents a scenario where y is entangled within parentheses and alongside other terms. Our primary objective is to isolate y on one side of the equation, thereby expressing its value in terms of the other variables, q and a. To achieve this, we will employ fundamental algebraic principles, such as the distributive property and the rearrangement of terms. Each step is a building block, contributing to the final solution. We'll start by expanding the left side of the equation, then strategically move terms around to group y terms together. This methodical approach ensures that we maintain the balance of the equation while progressing towards our goal. The process will involve addition, subtraction, and potentially division, all executed with precision to avoid errors. This section will serve as a roadmap, guiding you through each stage of the solution, ensuring that the logic and reasoning behind each step are crystal clear. By the end of this section, you'll have a solid grasp of the techniques required to tackle similar algebraic challenges.

Step 1: Distribute q on the Left Side

The first step in solving for y involves applying the distributive property to the left side of the equation, q(a+y) = 67y + 93. The distributive property dictates that we multiply the term outside the parentheses, which is q in this case, by each term inside the parentheses, which are a and y. This process effectively eliminates the parentheses, allowing us to work with individual terms. When we multiply q by a, we obtain qa, and when we multiply q by y, we get qy. Therefore, applying the distributive property transforms the left side of the equation from q(a+y) to qa + qy. The equation now reads qa + qy = 67y + 93. This seemingly simple step is crucial as it lays the groundwork for further simplification and rearrangement of the equation. By distributing q, we've freed up y from the confines of the parentheses, making it easier to isolate in subsequent steps. This step showcases the importance of understanding fundamental algebraic properties in manipulating equations and progressing towards a solution. The expanded form of the equation now presents a clearer picture of how y interacts with other terms, setting the stage for the next phase of the solution.

Step 2: Rearrange the Equation to Group y Terms

Having successfully distributed q, our next objective is to rearrange the equation qa + qy = 67y + 93 in such a way that all terms containing y are grouped on one side. This strategic maneuver is essential for isolating y and ultimately solving for its value. To achieve this, we need to identify the terms that include y, which are qy on the left side and 67y on the right side. Our goal is to consolidate these terms on a single side of the equation. A common approach is to subtract 67y from both sides of the equation. This action effectively eliminates 67y from the right side and introduces a -67y term on the left side. The equation then transforms into qa + qy - 67y = 93. Now, we have all the y terms on the left side. However, we also have the qa term on the left side, which does not contain y. To further isolate the y terms, we subtract qa from both sides of the equation. This action cancels out the qa term on the left side and introduces a -qa term on the right side. The equation now becomes qy - 67y = 93 - qa. This rearrangement is a critical step in our quest to solve for y, as it brings all the y terms together, paving the way for factoring and, ultimately, isolation. The equation is now in a form where we can clearly see the y terms and proceed with the next step in the solution.

Step 3: Factor out y

With the y terms now grouped together on one side of the equation (qy - 67y = 93 - qa), the next logical step is to factor out y. Factoring is a fundamental algebraic technique that allows us to express a sum or difference of terms as a product of a common factor and a resulting expression. In this case, y is the common factor in the terms qy and -67y. When we factor out y, we are essentially reversing the distributive property. We ask ourselves, what do we need to multiply y by to get qy, and what do we need to multiply y by to get -67y? The answers are q and -67, respectively. Therefore, factoring out y from the left side of the equation transforms qy - 67y into y(q - 67). The equation now reads y(q - 67) = 93 - qa. This step is a crucial turning point in our solution, as it isolates y as a single factor multiplied by an expression. The equation is now in a form that is very close to our desired outcome: expressing y in terms of q and a. Factoring out y simplifies the equation and brings us one step closer to the final solution. The next step will involve isolating y completely by eliminating the (q - 67) term.

Step 4: Isolate y by Dividing

Having successfully factored out y, we now have the equation in the form y(q - 67) = 93 - qa. The final step in solving for y is to isolate it completely. To achieve this, we need to eliminate the term (q - 67) that is multiplying y. The principle we employ here is the inverse operation: since y is being multiplied by (q - 67), we divide both sides of the equation by (q - 67). This action will cancel out the (q - 67) term on the left side, leaving y isolated. When we divide both sides of the equation by (q - 67), we obtain: [y(q - 67)] / (q - 67) = (93 - qa) / (q - 67). On the left side, the (q - 67) terms cancel each other out, leaving us with just y. On the right side, we have the expression (93 - qa) / (q - 67), which represents the value of y in terms of q and a. Therefore, the solution to the equation is y = (93 - qa) / (q - 67). This step completes our journey to solve for y. We have successfully isolated y and expressed it as a function of q and a. This final step demonstrates the power of inverse operations in solving algebraic equations and highlights the importance of maintaining balance on both sides of the equation throughout the solution process.

The Solution: y = (93 - qa) / (q - 67)

After meticulously navigating through the steps of distributing, rearranging, factoring, and dividing, we arrive at the solution for y in the equation q(a+y) = 67y + 93. The solution, elegantly expressed, is y = (93 - qa) / (q - 67). This equation reveals the value of y in terms of the parameters q and a. It signifies that for any given values of q and a (with the crucial caveat that q ≠ 67, as this would lead to division by zero), we can determine the corresponding value of y that satisfies the original equation. The journey to this solution underscores the fundamental principles of algebra, including the distributive property, the importance of rearranging terms, the power of factoring, and the necessity of maintaining balance when manipulating equations. This solution is not just a numerical answer; it's a testament to the logical and systematic approach to problem-solving that algebra provides. It's a culmination of careful steps, each building upon the previous, to unravel the relationship between y, q, and a. Understanding this solution is not just about memorizing a formula; it's about grasping the underlying process and applying it to other algebraic challenges. The solution y = (93 - qa) / (q - 67) stands as a concise and powerful expression of the value of y, a reward for our algebraic exploration.

Important Considerations and Restrictions

While the solution y = (93 - qa) / (q - 67) elegantly expresses y in terms of q and a, it's crucial to acknowledge certain considerations and restrictions that accompany this solution. These considerations stem from the inherent rules of mathematics and the specific structure of the equation. The most significant restriction arises from the denominator of the solution, (q - 67). In mathematics, division by zero is undefined, as it leads to mathematical inconsistencies and paradoxes. Therefore, the solution y = (93 - qa) / (q - 67) is valid only when the denominator (q - 67) is not equal to zero. This implies that q cannot be equal to 67. If q were to equal 67, the denominator would become zero, rendering the solution undefined and meaningless. This restriction highlights the importance of considering the domain of variables when solving equations. The solution is only valid within a specific range of values for q. Furthermore, it's worth noting that the solution provides a general expression for y. Depending on the specific values of q and a, y can take on a variety of values, both positive and negative. The solution encapsulates the relationship between y, q, and a, but it's the specific values of q and a that ultimately determine the numerical value of y. These considerations are essential for a complete understanding of the solution and its applicability. Ignoring these restrictions can lead to erroneous interpretations and applications of the solution. Therefore, a thorough understanding of the context and limitations of the solution is paramount.

Conclusion

In conclusion, the journey to solve the equation q(a+y) = 67y + 93 for y has been a rewarding exploration of algebraic principles and techniques. Through a step-by-step process of distribution, rearrangement, factoring, and division, we have successfully derived the solution y = (93 - qa) / (q - 67). This solution elegantly expresses y in terms of the parameters q and a, providing a general formula for determining the value of y given specific values for q and a. The process of solving this equation has reinforced the importance of fundamental algebraic concepts and the systematic approach required to tackle mathematical problems. We have also highlighted the crucial restriction that q ≠ 67, stemming from the prohibition of division by zero, underscoring the significance of considering the domain of variables in mathematical solutions. This exercise serves as a valuable illustration of how algebraic manipulation can unravel complex relationships and provide concise solutions. The solution y = (93 - qa) / (q - 67) is not just a final answer; it's a testament to the power of algebraic reasoning and the beauty of mathematical expression. By understanding the steps involved and the underlying principles, we can apply these skills to a wide range of mathematical challenges and appreciate the elegance and precision of algebraic solutions. This exploration has not only provided us with a solution to a specific equation but has also enhanced our understanding of the broader landscape of algebra and its problem-solving capabilities. The journey from the initial equation to the final solution is a testament to the power and elegance of mathematical reasoning.