Solving 2/(x-9) = 7/(4x+8) Step-by-Step Guide

Introduction

In this article, we will delve into the process of solving for x in the equation $\frac{2}{x-9}=\frac{7}{4x+8}$ and simplifying the answer fully. This type of problem is common in algebra and requires a solid understanding of fractions, cross-multiplication, and basic algebraic manipulation. We will walk through each step in detail, ensuring clarity and comprehension. Mastering these techniques is crucial for success in more advanced mathematical topics. The process involves several key steps, including cross-multiplication, distribution, combining like terms, isolating the variable, and finally, simplifying the solution. Each of these steps will be explained thoroughly to provide a comprehensive understanding of the solution.

Understanding the Equation

Before diving into the solution, it's essential to understand the equation we are working with: $\frac{2}{x-9}=\frac{7}{4x+8}$. This equation involves two fractions set equal to each other, with x appearing in the denominators. To solve for x, we need to eliminate the fractions. The most effective method for this is cross-multiplication. Cross-multiplication allows us to transform the equation into a more manageable linear form. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. This technique is based on the fundamental property that if two ratios are equal, their cross-products are also equal. Understanding this principle is crucial for solving various algebraic equations involving fractions. Additionally, it's important to note any restrictions on the value of x. The denominators cannot be zero, so we must ensure that our final solution does not make either $x-9$ or $4x+8$ equal to zero. This will be checked at the end of our calculations to confirm the validity of the solution.

Step-by-Step Solution

1. Cross-Multiplication

The first step in solving the equation $\frac{2}{x-9}=\frac{7}{4x+8}$ is to cross-multiply. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and the numerator of the second fraction by the denominator of the first fraction. This gives us:

$2(4x+8) = 7(x-9)$

Cross-multiplication is a fundamental technique for eliminating fractions in an equation. By multiplying across the equals sign, we transform the equation into a more manageable form that is easier to solve. This step is crucial because it removes the fractions, allowing us to work with a linear equation. The result of this step is a new equation that we can then simplify and solve for x. It's important to ensure that each term is multiplied correctly to avoid errors in the subsequent steps. This process is based on the property that if $\frac{a}{b}=\frac{c}{d}$, then $ad=bc$. This transformation is a key step in solving equations involving fractions and sets the stage for the following algebraic manipulations.

2. Distribution

Next, we need to distribute the constants on both sides of the equation. This involves multiplying the constants outside the parentheses by each term inside the parentheses:

$2(4x+8) = 2 * 4x + 2 * 8 = 8x + 16$

$7(x-9) = 7 * x - 7 * 9 = 7x - 63$

So, the equation becomes:

$8x + 16 = 7x - 63$

Distribution is a critical algebraic operation that involves multiplying a term outside a set of parentheses by each term inside the parentheses. This step is essential for expanding the equation and preparing it for further simplification. In this case, we distribute the 2 across $(4x + 8)$ and the 7 across $(x - 9)$. Accurate distribution is crucial for maintaining the equality of the equation and avoiding errors. The distributive property, which states that $a(b + c) = ab + ac$, is the foundation of this step. By correctly applying the distributive property, we transform the equation into a form where like terms can be combined and the variable x can be isolated. This step bridges the gap between the cross-multiplication result and the subsequent steps of combining like terms and solving for x.

3. Combining Like Terms

Now, we need to combine like terms to isolate x on one side of the equation. We can do this by subtracting $7x$ from both sides:

$8x + 16 - 7x = 7x - 63 - 7x$

$x + 16 = -63$

Next, subtract 16 from both sides:

$x + 16 - 16 = -63 - 16$

$x = -79$

Combining like terms is a fundamental algebraic technique used to simplify equations. This step involves grouping and combining terms that contain the same variable or are constants. By isolating the variable x on one side of the equation, we bring the equation closer to its solution. In this case, we first subtract $7x$ from both sides to move the x terms to the left side. Then, we subtract 16 from both sides to isolate x completely. This process relies on the properties of equality, ensuring that the equation remains balanced throughout the simplification. Accurate combination of like terms is crucial for arriving at the correct solution. This step reduces the complexity of the equation, making it easier to identify the value of x. The result is a simplified equation that directly reveals the solution for the variable.

4. Simplify the Answer

We have found that $x = -79$. Now, we need to check if this solution makes the denominators of the original equation equal to zero.

For the first denominator, $x - 9 = -79 - 9 = -88$, which is not zero.

For the second denominator, $4x + 8 = 4(-79) + 8 = -316 + 8 = -308$, which is also not zero.

Since neither denominator is zero, our solution is valid.

Therefore, the simplified answer is:

$x = -79$

Simplifying the answer is the final step in solving an algebraic equation. This involves ensuring that the solution is in its most basic form and that it satisfies the original equation's conditions. In this case, we have found that $x = -79$. However, it is crucial to verify that this solution does not make any of the denominators in the original equation equal to zero, as this would result in an undefined expression. We check the first denominator, $x - 9$, and find that when $x = -79$, it equals $-88$, which is not zero. Similarly, we check the second denominator, $4x + 8$, and find that when $x = -79$, it equals $-308$, which is also not zero. Since both denominators are non-zero, we can confidently conclude that our solution $x = -79$ is valid and fully simplified. This verification step is essential for ensuring the accuracy and completeness of the solution.

Conclusion

In conclusion, we have successfully solved the equation $\frac{2}{x-9}=\frac{7}{4x+8}$ for x and fully simplified the answer. The steps involved cross-multiplication, distribution, combining like terms, and verifying the solution. The final answer is $x = -79$. This process demonstrates the importance of understanding and applying fundamental algebraic techniques to solve equations effectively. Mastering these skills is essential for success in mathematics and related fields. Each step, from cross-multiplication to simplification, plays a crucial role in arriving at the correct solution. By following these steps meticulously, you can confidently solve similar equations and enhance your algebraic problem-solving abilities. This exercise not only provides a solution to a specific problem but also reinforces the underlying principles of algebraic manipulation and equation solving. The ability to solve such equations is a valuable skill that extends beyond the classroom and into various real-world applications.