Solving For Q In The Equation 2/(2q+3) = 1/8 A Step-by-Step Guide

In the realm of mathematics, solving for variables is a fundamental skill. This article delves into the step-by-step process of solving the equation $\frac{2}{2q+3} = \frac{1}{8}$ for the variable q. This type of equation, involving fractions and variables in the denominator, is common in algebra and requires a systematic approach to find the solution. Understanding how to solve such equations is crucial for various applications in mathematics, science, and engineering. We will break down each step, providing clear explanations and justifications to ensure a comprehensive understanding of the solution process. This article aims to not only provide the answer but also to enhance your problem-solving skills in algebra. The ability to manipulate equations and isolate variables is a cornerstone of mathematical proficiency, and this example serves as an excellent exercise in developing that skill.

The equation we aim to solve is $\frac{2}{2q+3} = \frac{1}{8}$. This is a rational equation, meaning it involves fractions with algebraic expressions. The variable q appears in the denominator of the fraction on the left side of the equation. To solve for q, we need to isolate it on one side of the equation. This involves a series of algebraic manipulations that preserve the equality. Our primary goal is to eliminate the fractions and simplify the equation into a more manageable form. This typically involves cross-multiplication, which is a technique used to eliminate denominators in equations involving fractions. By cross-multiplying, we transform the equation into a linear equation, which is much easier to solve. Understanding the structure of the equation and the properties of equality is crucial for successfully solving for q. Each step we take must be justified by mathematical principles, ensuring that the solution we obtain is accurate and valid. Before diving into the steps, it's important to recognize the potential restrictions on the variable q. Since the denominator cannot be zero, we need to be mindful of any values of q that would make $2q + 3 = 0$. This will help us avoid extraneous solutions. This preliminary analysis sets the stage for a methodical solution process.

To solve the equation $\frac{2}{2q+3} = \frac{1}{8}$, we begin by eliminating the fractions. This can be achieved through cross-multiplication. Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the results equal to each other. In this case, we multiply 2 (the numerator on the left side) by 8 (the denominator on the right side) and set it equal to 1 (the numerator on the right side) multiplied by $(2q + 3)$ (the denominator on the left side). This gives us the equation:

$2 * 8 = 1 * (2q + 3)$

Simplifying the left side, we get:

$16 = 1 * (2q + 3)$

Now, we simplify the right side by distributing the 1 across the parentheses:

$16 = 2q + 3$

Our next goal is to isolate the term with q on one side of the equation. To do this, we subtract 3 from both sides of the equation. This maintains the equality and moves the constant term to the left side:

$16 - 3 = 2q + 3 - 3$

Simplifying, we get:

$13 = 2q$

Finally, to solve for q, we divide both sides of the equation by 2:

$\frac{13}{2} = \frac{2q}{2}$

This simplifies to:

$q = \frac{13}{2}$

Thus, the solution to the equation $\frac{2}{2q+3} = \frac{1}{8}$ is $q = \frac{13}{2}$. We have systematically isolated q by using cross-multiplication and applying the properties of equality.

To ensure the accuracy of our solution, it's crucial to verify that $q = \frac{13}{2}$ indeed satisfies the original equation. This step is vital to prevent errors and confirm the correctness of our algebraic manipulations. We substitute $q = \frac{13}{2}$ back into the original equation $\frac{2}{2q+3} = \frac{1}{8}$ and check if the left side equals the right side.

Substituting $q = \frac{13}{2}$, we get:

$\frac{2}{2(\frac{13}{2})+3} = \frac{1}{8}$

First, we simplify the denominator:

$2(\frac{13}{2}) = 13$

So the denominator becomes:

$13 + 3 = 16$

Now, we substitute this back into the fraction:

$\frac{2}{16} = \frac{1}{8}$

Simplifying the fraction on the left side, we get:

$\frac{2}{16} = \frac{1}{8}$

Thus, the left side of the equation equals the right side:

$\frac{1}{8} = \frac{1}{8}$

This confirms that our solution $q = \frac{13}{2}$ is correct. By substituting the solution back into the original equation and verifying that it holds true, we have provided a rigorous check of our work. This process not only ensures the correctness of the solution but also reinforces the understanding of the algebraic steps involved.

When solving equations like $\frac{2}{2q+3} = \frac{1}{8}$, several common mistakes can occur. Recognizing these pitfalls and understanding how to avoid them is crucial for achieving accurate solutions. One frequent error is incorrect cross-multiplication. For instance, students might multiply only one term in the denominator or mix up the numerators and denominators. To prevent this, always ensure that you multiply the numerator of each fraction by the denominator of the other fraction. Write out each step clearly to avoid errors in multiplication and distribution.

Another common mistake involves the order of operations. When simplifying expressions, remember to follow the correct order (PEMDAS/BODMAS). In our case, it's essential to perform the multiplication in the denominator $(2 * \frac{13}{2})$ before adding 3. Skipping or misapplying this order can lead to incorrect results. Always double-check the order in which you perform operations.

Sign errors are also a frequent source of mistakes, especially when dealing with negative numbers or subtracting terms. For example, when subtracting 3 from both sides of the equation, ensure that you apply the subtraction correctly to both sides. A careless sign error can completely alter the solution. Take your time and pay close attention to the signs of the terms.

Finally, neglecting to verify the solution is a significant oversight. Always substitute your solution back into the original equation to check for correctness. This step can reveal errors in your calculations or algebraic manipulations. By avoiding these common mistakes and adopting careful, methodical problem-solving techniques, you can significantly improve your accuracy and confidence in solving algebraic equations.

While cross-multiplication is a standard method for solving equations like $\frac{2}{2q+3} = \frac{1}{8}$, there are alternative approaches that can be equally effective. Understanding these different methods can enhance your problem-solving flexibility and provide a deeper understanding of algebraic principles. One alternative method involves multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the denominators are $(2q + 3)$ and 8. The LCM is $8(2q + 3)$. By multiplying both sides of the equation by this LCM, we eliminate the fractions in one step:

$8(2q + 3) * \frac{2}{2q+3} = 8(2q + 3) * \frac{1}{8}$

On the left side, $(2q + 3)$ cancels out, and on the right side, 8 cancels out. This simplifies the equation to:

$8 * 2 = (2q + 3)$

$16 = 2q + 3$

From this point, the solution proceeds as before, subtracting 3 from both sides and then dividing by 2 to find $q = \frac{13}{2}$. This method avoids the direct cross-multiplication step but still effectively clears the fractions.

Another approach is to take the reciprocal of both sides of the equation. The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$. Taking the reciprocal of both sides of $\frac{2}{2q+3} = \frac{1}{8}$ gives us:

$\frac{2q+3}{2} = 8$

Now, we can multiply both sides by 2 to eliminate the fraction:

$2q + 3 = 16$

Again, the solution proceeds as before, leading to $q = \frac{13}{2}$. This method is particularly useful when the variable is in the denominator and taking reciprocals simplifies the equation structure. By exploring these alternative methods, you can gain a more comprehensive understanding of algebraic techniques and choose the method that best suits the problem at hand.

In conclusion, we have thoroughly explored the process of solving the equation $\frac{2}{2q+3} = \frac{1}{8}$ for the variable q. We began by understanding the structure of the equation and recognizing it as a rational equation. We then systematically applied cross-multiplication to eliminate the fractions and simplify the equation. Each step was carefully explained, ensuring clarity and precision in our algebraic manipulations. We isolated the variable q and arrived at the solution $q = \frac{13}{2}$. To guarantee the accuracy of our solution, we performed a verification step by substituting $q = \frac{13}{2}$ back into the original equation. This confirmed that our solution was indeed correct. Furthermore, we discussed common mistakes that students often make when solving such equations and provided strategies to avoid them. These included errors in cross-multiplication, order of operations, sign errors, and neglecting to verify the solution.

Additionally, we explored alternative methods for solving the equation, such as multiplying both sides by the least common multiple of the denominators and taking the reciprocal of both sides. These alternative approaches provide valuable insights into different algebraic techniques and enhance problem-solving flexibility. By mastering these methods, you can approach a wide range of algebraic problems with greater confidence and proficiency. The ability to solve equations like this is a fundamental skill in mathematics and has applications in various fields, including science, engineering, and economics. This comprehensive exploration of the solution process not only provides the answer but also aims to deepen your understanding of algebraic principles and problem-solving strategies.