Solving And Verifying The Equation: 14/(x+1) = (x+6)/(x+1)

Solving equations can seem daunting, but breaking it down into manageable steps makes it much easier. In this guide, we'll tackle the equation ${\frac{14}{x+1}=\frac{x+6}{x+1}}$. We'll go through each step meticulously, ensuring you not only understand how to solve it but also how to verify your solution. Understanding the methods to solve equations is a crucial skill in mathematics, forming the basis for more advanced topics. This particular equation involves fractions and algebraic expressions, providing a good exercise in handling these elements. By mastering these techniques, you’ll be better prepared for more complex mathematical challenges. So, let's dive in and demystify this equation together. Remember, the key is to take it one step at a time and ensure you understand the logic behind each operation.

1. Understanding the Equation

At first glance, the equation ${\frac{14}{x+1}=\frac{x+6}{x+1}}$ might seem a bit intimidating. But don't worry! The first step in solving any equation is to understand its structure. We have a rational equation here, where both sides are fractions with algebraic expressions. Specifically, we have two fractions that are equal to each other. Both fractions have the same denominator, which is x + 1. This common denominator is a key observation that simplifies our solving process significantly. Before we jump into the algebraic manipulations, let's consider what this means. The equation essentially states that the quotient of 14 divided by (x + 1) is equal to the quotient of (x + 6) divided by (x + 1). This understanding is crucial because it helps us strategize our approach. Knowing that the denominators are the same gives us a direct path to equating the numerators, but we need to be cautious about potential restrictions on x. Specifically, the denominator cannot be zero, which we will address later. Recognizing the structure of the equation is half the battle. By understanding that we're dealing with a rational equation with a common denominator, we can proceed confidently to the next steps. Always take a moment to analyze the equation before you start manipulating it – it can save you a lot of time and effort.

2. Clearing the Denominator

Now that we understand our equation ${\frac{14}{x+1}=\frac{x+6}{x+1}}$, the next step is to eliminate the denominators. This simplifies the equation and makes it easier to solve. Since both fractions have the same denominator, (x + 1), we can multiply both sides of the equation by this common denominator. By multiplying both sides by (x + 1), we effectively cancel out the denominators. This is a valid operation as long as x + 1 is not equal to zero. Multiplying both sides of the equation by (x + 1) gives us:

$$(x+1) \cdot \frac{14}{x+1} = (x+1) \cdot \frac{x+6}{x+1}$$

Notice how (x + 1) appears in both the numerator and the denominator on each side, allowing us to cancel them out. After canceling, the equation simplifies significantly. This simplification is a crucial step because it transforms a complex rational equation into a much simpler linear equation. This is a common technique in algebra – reducing complex expressions into simpler forms that are easier to manipulate. By clearing the denominator, we pave the way for the next steps in solving for x. However, it's important to remember our initial restriction that x + 1 cannot be zero, which means x cannot be -1. We’ll keep this in mind as we proceed and verify our solution later. So, with the denominators cleared, we’re one step closer to finding the value of x.

3. Simplifying the Equation

After clearing the denominator in the equation ${\frac{14}{x+1}=\frac{x+6}{x+1}}$, we arrive at a much simpler form. Cancelling the (x + 1) terms from both sides leaves us with:

$$14 = x + 6$$

This is a linear equation, which is significantly easier to solve than the original rational equation. The goal now is to isolate x on one side of the equation. To do this, we need to get rid of the + 6 on the right side. We can achieve this by subtracting 6 from both sides of the equation. Remember, whatever operation you perform on one side of the equation, you must also perform on the other side to maintain the equality. Subtracting 6 from both sides gives us:

$$14 - 6 = x + 6 - 6$$

Now, we simplify each side of the equation. On the left side, 14 - 6 equals 8. On the right side, + 6 and - 6 cancel each other out, leaving us with just x. This simplification is a fundamental algebraic step, making it clear what the value of x is. The process of simplifying equations is essential in mathematics. It allows us to break down complex problems into manageable parts. By isolating the variable, we can directly see its value, which is the ultimate goal of solving an equation. With this simplification, we're very close to our solution. We've successfully transformed the original equation into a form where the value of x is almost immediately apparent.

4. Solving for x

Having simplified the equation ${\frac{14}{x+1}=\frac{x+6}{x+1}}$ to 14 = x + 6, we're now in the final stage of solving for x. As we established in the previous step, subtracting 6 from both sides of the equation isolates x. This gives us:

$$14 - 6 = x + 6 - 6$$

Simplifying this, we find:

$$8 = x$$

So, the solution to the equation is x = 8. This is a critical point, but we're not quite done yet. While we've found a potential solution, we must verify that it satisfies the original equation and adheres to any restrictions we identified earlier. Remember, when we cleared the denominator, we noted that x cannot be -1, because that would make the denominator zero, which is undefined. Our solution, x = 8, does not violate this restriction, so it's a promising candidate. Solving for x is often the primary objective in algebra, but it’s equally important to ensure the solution is valid within the context of the original problem. By isolating x, we've determined its value, but the next step—verification—is crucial to confirm the accuracy and applicability of our solution. This process of solving and verifying is a cornerstone of mathematical problem-solving, ensuring we arrive at correct and meaningful answers. With x = 8 in hand, let's proceed to the verification step to finalize our solution.

5. Verifying the Solution

Now that we've solved the equation ${\frac{14}{x+1}=\frac{x+6}{x+1}}$ and found x = 8, the most crucial step remains: verifying the solution. This ensures that our value for x not only satisfies the simplified equation but also the original equation, and that it doesn't violate any restrictions. To verify, we substitute x = 8 back into the original equation:

$$\frac{14}{x+1} = \frac{x+6}{x+1}$$

Substitute x = 8:

$$\frac{14}{8+1} = \frac{8+6}{8+1}$$

Now, simplify both sides of the equation. On the left side, 8 + 1 is 9, so we have \frac{14}{9}. On the right side, 8 + 6 is 14, and 8 + 1 is 9, so we have \frac{14}{9}. Thus, the equation becomes:

$$\frac{14}{9} = \frac{14}{9}$$

This statement is true, which confirms that x = 8 is indeed a valid solution. Verification is an indispensable part of the problem-solving process, especially in equations involving fractions or radicals, where extraneous solutions can arise. By substituting the solution back into the original equation, we guarantee its correctness. This step also reinforces our understanding of the equation and the solution's place within it. Moreover, we recall that our initial restriction was x ≠ -1, which x = 8 satisfies. Therefore, we can confidently conclude that x = 8 is the correct and valid solution to the equation. This thorough verification process underscores the importance of precision and attention to detail in mathematics.

Conclusion

In this comprehensive guide, we've successfully solved the equation ${\frac{14}{x+1}=\frac{x+6}{x+1}}$ and verified our solution. We began by understanding the structure of the equation, which helped us strategize our approach. We then cleared the denominator, simplified the equation, and solved for x, finding that x = 8. Crucially, we verified our solution by substituting it back into the original equation, confirming its validity. This step-by-step process illustrates the importance of a systematic approach to solving algebraic equations. Each step, from understanding the equation to verifying the solution, plays a vital role in ensuring accuracy and understanding. The techniques we've employed here, such as clearing denominators and simplifying, are fundamental in algebra and can be applied to a wide range of problems. Remember, solving equations is not just about finding the answer; it's about understanding the process and the logic behind each step. By mastering these skills, you’ll be well-equipped to tackle more complex mathematical challenges. And always remember, verification is key to ensuring the correctness of your solutions. Continue practicing and applying these techniques, and you'll find yourself becoming more confident and proficient in solving equations. For further learning and resources, consider exploring websites like Khan Academy's Algebra Section, which offers extensive lessons and practice problems on various algebraic topics.