Solving The Equation 9/(7x+28) = 9/(x+4) - 3/7 A Step By Step Guide

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Introduction

In this article, we will delve into the process of solving the equation $\frac{9}{7x+28} = \frac{9}{x+4} - \frac{3}{7}$. This type of equation involves rational expressions, and finding its solutions requires a systematic approach. We will break down each step, providing a clear and detailed explanation to ensure a comprehensive understanding. Whether you are a student tackling algebra problems or someone looking to refresh your math skills, this guide will help you master the techniques needed to solve similar equations. The key to solving such problems lies in manipulating the equation to isolate the variable, which in this case is x. This involves clearing fractions, combining like terms, and applying algebraic principles to arrive at the solution. We will cover each of these aspects in detail, making sure that you grasp the underlying concepts and can apply them confidently.

Step-by-Step Solution

1. Simplify the Equation

To begin, our primary objective is to simplify the given equation: $\frac{9}{7x+28} = \frac{9}{x+4} - \frac{3}{7}$. The first step involves factoring the denominator on the left side. We notice that $7x + 28$ can be factored as $7(x+4)$. This simplifies our equation to:

$\frac{9}{7(x+4)} = \frac{9}{x+4} - \frac{3}{7}$

This initial simplification is crucial because it allows us to identify common factors and streamline the subsequent steps. Factoring the denominator helps in recognizing the common terms, which will be beneficial when we clear the fractions later on. Furthermore, this step sets the stage for identifying any restrictions on the variable x, which we will address shortly.

2. Identify Restrictions on x

Before proceeding further, it’s crucial to identify any values of x that would make the denominators in our equation equal to zero. These values are restrictions on x because division by zero is undefined. Looking at the denominators in the equation $\frac{9}{7(x+4)} = \frac{9}{x+4} - \frac{3}{7}$, we see that $x + 4$ appears in two of the denominators. Setting $x + 4 = 0$ gives us $x = -4$. Therefore, $x$ cannot be equal to $-4$, as this would make the denominators zero and invalidate the equation. Identifying restrictions early on ensures that we avoid extraneous solutions later in the process. This step is a fundamental aspect of solving rational equations and helps maintain the integrity of our solution.

3. Clear the Fractions

To eliminate the fractions, we need to find the least common denominator (LCD) of the terms in the equation. The denominators are $7(x+4)$, $(x+4)$, and $7$. The LCD is the smallest expression that is divisible by each of these denominators, which in this case is $7(x+4)$. We multiply both sides of the equation by the LCD to clear the fractions:

$7(x+4) \cdot \frac{9}{7(x+4)} = 7(x+4) \cdot \left(\frac{9}{x+4} - \frac{3}{7}\right)$

This step is crucial because it transforms the equation from one involving fractions to a simpler, more manageable form. By multiplying each term by the LCD, we effectively eliminate the denominators, making the equation easier to solve. The distribution of the LCD across the terms on both sides ensures that the equation remains balanced and equivalent to the original.

4. Distribute and Simplify

Now, we distribute the $7(x+4)$ on the right side of the equation and simplify:

$9 = 7(x+4) \cdot \frac{9}{x+4} - 7(x+4) \cdot \frac{3}{7}$

Simplifying each term, we get:

$9 = 7 \cdot 9 - (x+4) \cdot 3$

$9 = 63 - 3(x+4)$

This step involves basic arithmetic operations, but it's essential to perform them accurately. Distributing and simplifying correctly ensures that the equation is transformed into a linear equation that can be solved easily. Attention to detail during this step is vital to avoid errors that could lead to an incorrect solution. The goal is to reduce the complexity of the equation by eliminating parentheses and combining like terms.

5. Further Simplification

Next, we continue to simplify the equation by distributing the $-3$ across $(x+4)$:

$9 = 63 - 3x - 12$

Combine the constant terms on the right side:

$9 = 51 - 3x$

This step is a continuation of the simplification process, aiming to isolate the variable term. Distributing and combining like terms helps in reducing the equation to a standard linear form, which is easier to solve. Accuracy in these steps is paramount to ensure the correctness of the final solution.

6. Isolate the Variable

To isolate the term with x, we subtract $51$ from both sides of the equation:

$9 - 51 = -3x$

$-42 = -3x$

Isolating the variable is a critical step in solving any equation. It involves performing operations on both sides to bring all terms involving the variable to one side and all constant terms to the other side. This step sets the stage for the final operation that will give us the value of x.

7. Solve for x

Finally, we divide both sides by $-3$ to solve for x:

$\frac{-42}{-3} = x$

$x = 14$

This final step gives us the solution for x. Dividing both sides by the coefficient of x isolates the variable and provides its value. It’s the culmination of all the previous steps and provides the answer to the equation.

8. Check the Solution

It’s essential to check our solution by substituting $x = 14$ back into the original equation:

$\frac{9}{7(14)+28} = \frac{9}{14+4} - \frac{3}{7}$

$\frac{9}{98+28} = \frac{9}{18} - \frac{3}{7}$

$\frac{9}{126} = \frac{1}{2} - \frac{3}{7}$

Simplify the fractions:

$\frac{1}{14} = \frac{7}{14} - \frac{6}{14}$

$\frac{1}{14} = \frac{1}{14}$

The solution checks out, as both sides of the equation are equal. This step is a crucial verification process to ensure that the solution we found is correct and that no errors were made during the solving process.

Final Answer

A. The solution is

$x = 14$

This is the final answer to the given equation. We have shown a step-by-step solution, explaining each part of the process to ensure a clear understanding of how to solve such equations. This approach not only provides the correct answer but also reinforces the underlying mathematical principles involved. Remember, practice is key to mastering these types of problems, and consistent effort will build your confidence and skills.

Conclusion

In conclusion, we have successfully solved the equation $\frac{9}{7x+28} = \frac{9}{x+4} - \frac{3}{7}$ by following a systematic approach. We began by simplifying the equation, identifying restrictions on x, clearing fractions, and then isolating the variable to find the solution. Our step-by-step method ensured clarity and accuracy, leading us to the final answer of $x = 14$. Remember, when tackling rational equations, it's crucial to be meticulous and check your solutions to avoid extraneous results. This comprehensive guide not only provides the solution but also equips you with the skills to approach similar problems confidently. Keep practicing, and you'll become proficient in solving various types of algebraic equations. Solving equations is a fundamental skill in mathematics, and mastering it opens the door to more advanced topics. By understanding the principles and techniques discussed here, you'll be well-prepared to tackle any equation that comes your way. Keep learning, keep practicing, and keep solving!