What Is The Sum Of $\frac{6x}{3x+4} + \frac{7}{3x+4}$?A. $\frac{6x+7}{(3x+4)^2}, \quad X \neq -\frac{4}{3}$ B. $\frac{13x}{3x+4}, \quad X \neq -\frac{4}{3}$ C. $\frac{42x}{3x+4}, \quad X \neq -\frac{4}{3}$ D.

Introduction

When dealing with fractions, one of the most common operations is addition. However, adding fractions can be a bit tricky, especially when they have different denominators. In such cases, we need to find a common denominator to add the fractions. But what if the fractions already have a common denominator? In this case, we can simply add the numerators and keep the denominator the same. In this article, we will explore the sum of two fractions with a common denominator, specifically the sum of $\frac{6x}{3x+4} + \frac{7}{3x+4}$.

Understanding the Problem

The problem asks us to find the sum of two fractions, $\frac{6x}{3x+4}$ and $\frac{7}{3x+4}$. The first fraction has a numerator of $6x$ and a denominator of $3x+4$, while the second fraction has a numerator of $7$ and the same denominator of $3x+4$. Since both fractions have the same denominator, we can add them directly.

Adding the Fractions

To add the fractions, we simply add the numerators and keep the denominator the same. This means that the sum of the two fractions is:

$$\frac{6x}{3x+4} + \frac{7}{3x+4} = \frac{6x+7}{3x+4}$$

Simplifying the Expression

However, we are not done yet. The expression $\frac{6x+7}{3x+4}$ can be simplified further. To do this, we can factor out the greatest common factor (GCF) of the numerator and denominator. In this case, the GCF is $1$, but we can still simplify the expression by dividing both the numerator and denominator by their greatest common factor.

Finding the Greatest Common Factor

To find the greatest common factor of $6x+7$ and $3x+4$, we can list the factors of each expression and find the greatest common factor. The factors of $6x+7$ are $1, 7, 6x+7$, and the factors of $3x+4$ are $1, 3x+4$. Since there are no common factors between the two expressions, the greatest common factor is $1$.

Simplifying the Expression Further

Since the greatest common factor is $1$, we cannot simplify the expression further by dividing both the numerator and denominator by their greatest common factor. However, we can still simplify the expression by factoring out the greatest common factor of the numerator and denominator.

Factoring Out the Greatest Common Factor

To factor out the greatest common factor of the numerator and denominator, we can divide both the numerator and denominator by their greatest common factor. In this case, the greatest common factor is $1$, so we cannot factor out any common factors.

The Final Answer

However, we can still simplify the expression by canceling out any common factors between the numerator and denominator. In this case, there are no common factors between the numerator and denominator, so we cannot cancel out any common factors.

Conclusion

In conclusion, the sum of $\frac{6x}{3x+4} + \frac{7}{3x+4}$ is $\frac{6x+7}{3x+4}$. This expression cannot be simplified further by factoring out the greatest common factor or canceling out any common factors between the numerator and denominator.

Final Answer

The final answer is $\boxed{\frac{6x+7}{3x+4}}$.

Discussion

The discussion category for this problem is mathematics. The problem involves adding two fractions with a common denominator and simplifying the resulting expression.

Related Problems

Some related problems to this one include:

• Adding two fractions with different denominators
• Simplifying an expression by factoring out the greatest common factor
• Canceling out common factors between the numerator and denominator

Example Problems

Some example problems to this one include:

• $\frac{2x}{x+1} + \frac{3}{x+1}$
• $\frac{4x}{2x+1} + \frac{5}{2x+1}$
• $\frac{6x}{3x+2} + \frac{7}{3x+2}$

Solutions

The solutions to these example problems are:

• $\frac{2x+3}{x+1}$
• $\frac{4x+5}{2x+1}$
• $\frac{6x+7}{3x+2}$
  

Introduction

In our previous article, we explored the sum of two fractions with a common denominator, specifically the sum of $\frac{6x}{3x+4} + \frac{7}{3x+4}$. We found that the sum of the two fractions is $\frac{6x+7}{3x+4}$. In this article, we will answer some frequently asked questions (FAQs) related to this problem.

Q&A

Q: What is the common denominator of the two fractions?

A: The common denominator of the two fractions is $3x+4$.

Q: How do I add two fractions with a common denominator?

A: To add two fractions with a common denominator, you simply add the numerators and keep the denominator the same.

Q: Can I simplify the expression $\frac{6x+7}{3x+4}$ further?

A: No, the expression $\frac{6x+7}{3x+4}$ cannot be simplified further by factoring out the greatest common factor or canceling out any common factors between the numerator and denominator.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{\frac{6x+7}{3x+4}}$.

Q: What are some related problems to this one?

A: Some related problems to this one include:

• Adding two fractions with different denominators
• Simplifying an expression by factoring out the greatest common factor
• Canceling out common factors between the numerator and denominator

Q: What are some example problems to this one?

A: Some example problems to this one include:

• $\frac{2x}{x+1} + \frac{3}{x+1}$
• $\frac{4x}{2x+1} + \frac{5}{2x+1}$
• $\frac{6x}{3x+2} + \frac{7}{3x+2}$

Q: What are the solutions to these example problems?

A: The solutions to these example problems are:

• $\frac{2x+3}{x+1}$
• $\frac{4x+5}{2x+1}$
• $\frac{6x+7}{3x+2}$

Conclusion

In conclusion, we have answered some frequently asked questions related to the sum of two fractions with a common denominator. We have also provided some example problems and their solutions. We hope that this article has been helpful in understanding the concept of adding fractions with a common denominator.

Final Answer

The final answer to the problem is $\boxed{\frac{6x+7}{3x+4}}$.

Discussion

The discussion category for this problem is mathematics. The problem involves adding two fractions with a common denominator and simplifying the resulting expression.

Related Problems

Some related problems to this one include:

• Adding two fractions with different denominators
• Simplifying an expression by factoring out the greatest common factor
• Canceling out common factors between the numerator and denominator

Example Problems

Some example problems to this one include:

• $\frac{2x}{x+1} + \frac{3}{x+1}$
• $\frac{4x}{2x+1} + \frac{5}{2x+1}$
• $\frac{6x}{3x+2} + \frac{7}{3x+2}$

Solutions

The solutions to these example problems are:

• $\frac{2x+3}{x+1}$
• $\frac{4x+5}{2x+1}$
• $\frac{6x+7}{3x+2}$