Solving For Q In The Equation 10/(2q+8) = 10 A Step-by-Step Guide

In this comprehensive article, we will delve into the step-by-step process of solving for the variable q in the equation 10/(2q+8) = 10. This equation falls under the realm of algebraic equations, where our primary goal is to isolate the variable q on one side of the equation to determine its value. Mastering the techniques to solve such equations is fundamental in mathematics, as they appear in various contexts, from basic algebra to more advanced calculus and physics problems. This guide is structured to provide a clear, methodical approach, ensuring that readers of all levels can follow along and understand each step. We will begin by outlining the initial equation and then proceed through each algebraic manipulation required to isolate q. This includes dealing with fractions, performing inverse operations, and simplifying the equation to arrive at the final solution. By the end of this article, you will not only understand how to solve this specific equation but also gain a broader understanding of how to approach similar algebraic problems. Let's embark on this mathematical journey together, breaking down each step and clarifying the underlying principles, to confidently find the value of q.

Step 1: Understand the Initial Equation

At the heart of our mathematical quest lies the equation: 10/(2q+8) = 10. This equation is a classic example of a rational equation, where a variable appears in the denominator of a fraction. Before we begin our algebraic manipulations, it's crucial to understand what this equation is telling us. We have a fraction, 10 divided by the expression (2q+8), and the result of this division is equal to 10. Our objective is clear: we need to find the value of q that makes this statement true. To do this, we must isolate q on one side of the equation. This involves carefully unwinding the operations that are being performed on q, one step at a time. We will use the fundamental principles of algebra, ensuring that any operation we perform on one side of the equation is also performed on the other side, maintaining the balance and integrity of the equation. Understanding the initial equation is more than just seeing the symbols; it's about grasping the relationship between the numbers and the variable, and how they interact. This understanding forms the bedrock of our solution process. By taking the time to truly comprehend the equation, we set ourselves up for success in solving for q. The subsequent steps will build upon this foundation, leading us to the final answer through a series of logical and methodical manipulations. Remember, each step is a deliberate action, aimed at bringing us closer to the solution, and a solid understanding of the equation is our compass.

Step 2: Eliminate the Fraction by Multiplying

To effectively solve for q in the equation 10/(2q+8) = 10, our initial hurdle is the fraction. Fractions can often complicate algebraic manipulations, so our first tactical move is to eliminate it. The most direct way to achieve this is by multiplying both sides of the equation by the denominator of the fraction, which in this case is (2q+8). This is a pivotal step because it clears the way for simpler algebraic operations. When we multiply the left side of the equation, 10/(2q+8), by (2q+8), the denominator cancels out, leaving us with just 10. On the right side of the equation, we multiply 10 by (2q+8), resulting in 10(2q+8). This multiplication maintains the balance of the equation, a fundamental principle in algebra. Our equation now transforms from 10/(2q+8) = 10 to 10 = 10(2q+8). This new form is significantly easier to work with, as we've successfully removed the fractional component. The next phase involves distributing the 10 on the right side, which will further simplify the equation and bring us closer to isolating q. By eliminating the fraction, we've taken a major step towards solving for q, paving the way for subsequent algebraic maneuvers that will lead us to the final solution. This process underscores the importance of strategic simplification in equation solving, a skill that is invaluable in mathematics.

Step 3: Distribute on the Right Side

Having successfully eliminated the fraction in our equation, the next crucial step in solving for q is to simplify the right side of the equation. Our equation currently stands as 10 = 10(2q+8). The presence of parentheses on the right side indicates that we need to apply the distributive property. This property allows us to multiply the term outside the parentheses, which is 10, by each term inside the parentheses, namely 2q and 8. When we distribute the 10, we perform the following operations: 10 * 2q and 10 * 8. This results in 20q and 80, respectively. So, the right side of the equation, 10(2q+8), transforms into 20q + 80. Our equation now looks like this: 10 = 20q + 80. This transformation is significant because it removes the parentheses, allowing us to further isolate q. The equation is now in a more manageable form, with all terms explicitly written out. The next step in our journey to solve for q involves isolating the term containing q, which is 20q. This will require us to perform additional algebraic operations, but with each step, we are getting closer to our goal. By distributing and simplifying, we continue to refine the equation, making it easier to unravel the value of q. This methodical approach is key to solving algebraic equations efficiently and accurately.

Step 4: Isolate the Term with q

With our equation now simplified to 10 = 20q + 80, the next strategic move in solving for q is to isolate the term that contains q, which is 20q. To achieve this, we need to eliminate the constant term that is added to 20q, which in this case is 80. We can do this by applying the inverse operation. Since 80 is being added, the inverse operation is subtraction. We subtract 80 from both sides of the equation to maintain balance. Subtracting 80 from the left side of the equation, 10, gives us 10 - 80, which equals -70. On the right side of the equation, subtracting 80 from 20q + 80 cancels out the 80, leaving us with just 20q. Our equation now transforms to -70 = 20q. This is a crucial step because we have successfully isolated the term containing q on one side of the equation. The remaining step to solve for q is to isolate q itself. This will involve dividing both sides of the equation by the coefficient of q, which is 20. By isolating the term with q, we've significantly narrowed down our focus, making the final step of solving for q more straightforward. This process underscores the importance of using inverse operations to strategically isolate variables in algebraic equations.

Step 5: Solve for q by Dividing

Having successfully isolated the term 20q in our equation, which now reads -70 = 20q, the final and decisive step in solving for q is to isolate q itself. Currently, q is being multiplied by 20. To undo this multiplication and isolate q, we employ the inverse operation, which is division. We divide both sides of the equation by 20 to maintain the equation's balance. On the left side of the equation, we have -70 divided by 20, which can be written as -70/20. This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 10. This simplification yields -7/2. On the right side of the equation, dividing 20q by 20 results in q. Therefore, our equation now transforms to q = -7/2. This is the solution to our equation. We have successfully isolated q and determined its value. The value of q that satisfies the original equation 10/(2q+8) = 10 is -7/2. This final step demonstrates the power of using inverse operations to unravel equations and isolate variables. By systematically applying these principles, we can confidently solve a wide range of algebraic problems. The solution q = -7/2 completes our journey, providing a definitive answer to the equation we set out to solve.

Step 6: Verification of the Solution

After diligently solving for q and arriving at the solution q = -7/2, the crucial final step is to verify our solution. This process involves substituting the value we found for q back into the original equation to ensure that it holds true. Verification is not just a formality; it's a critical safeguard against errors that may have occurred during the algebraic manipulations. It provides us with the assurance that our solution is correct and satisfies the initial conditions of the equation. Our original equation is 10/(2q+8) = 10. To verify, we substitute q with -7/2: 10/(2(-7/2)+8) = 10 First, we simplify the expression inside the parentheses in the denominator. Multiplying 2 by -7/2 gives us -7. So, the denominator becomes -7 + 8, which simplifies to 1. Our equation now looks like this: 10/1 = 10. Dividing 10 by 1 indeed equals 10, so the equation holds true. This verification confirms that our solution q = -7/2 is correct. The process of verification underscores the importance of precision in mathematics. It reinforces the idea that solving an equation is not just about finding a value; it's about ensuring that the value is accurate and consistent with the given conditions. By verifying our solution, we complete the problem-solving process with confidence, knowing that we have arrived at the correct answer. This step is a testament to the rigor and thoroughness required in mathematical problem-solving.

In conclusion, solving for q in the equation 10/(2q+8) = 10 has been a methodical journey through the realms of algebraic manipulation. We began by understanding the equation, then strategically eliminated the fraction, simplified the equation through distribution, isolated the term containing q, and finally, isolated q itself to arrive at the solution q = -7/2. The journey culminated in a critical verification step, where we confirmed the accuracy of our solution by substituting it back into the original equation. This process exemplifies the core principles of algebraic problem-solving: understanding the problem, strategically planning the steps, executing those steps with precision, and verifying the solution to ensure accuracy. Each step in the process, from eliminating the fraction to distributing and isolating terms, highlights the importance of inverse operations and maintaining the balance of the equation. The successful solution of this equation not only provides the value of q but also reinforces the broader skills of algebraic manipulation, logical thinking, and problem-solving. These skills are fundamental in mathematics and have wide-ranging applications in various fields. By mastering these techniques, one can confidently approach a multitude of algebraic problems and develop a deeper appreciation for the elegance and power of mathematical reasoning. The journey to solve for q is a microcosm of the larger mathematical journey, where each problem solved is a step forward in understanding the intricate patterns and relationships that govern the world around us. This exploration solidifies the foundation for tackling more complex mathematical challenges in the future.