To solve the problem of combining the fractions [tex]\(\frac{6}{x} + \frac{5}{7}\)[/tex] into a single expression, follow these steps:
1. Identify a Common Denominator: The denominators in the given fractions are [tex]\(x\)[/tex] and 7. To combine these fractions, we need a common denominator, which would be the product of the individual denominators. Therefore, the common denominator is [tex]\(7x\)[/tex].
2. Rewrite Each Fraction with the Common Denominator:
- For [tex]\(\frac{6}{x}\)[/tex], multiply both the numerator and denominator by 7 to get [tex]\(\frac{6 \cdot 7}{x \cdot 7} = \frac{42}{7x}\)[/tex].
- For [tex]\(\frac{5}{7}\)[/tex], multiply both the numerator and denominator by [tex]\(x\)[/tex] to get [tex]\(\frac{5 \cdot x}{7 \cdot x} = \frac{5x}{7x}\)[/tex].
3. Combine the Fractions Using the Common Denominator:
Once the fractions are rewritten to share a common denominator, they can be added directly:
[tex]\[
\frac{42}{7x} + \frac{5x}{7x} = \frac{42 + 5x}{7x}
\][/tex]
Therefore, the combined fraction [tex]\(\frac{6}{x} + \frac{5}{7}\)[/tex] can be simplified to:
[tex]\(\frac{42 + 5x}{7x}\)[/tex].
Comparing this with the given options:
- F. [tex]\(\frac{11}{7 x}\)[/tex]
- G. [tex]\(\frac{30}{7 x}\)[/tex]
- H. [tex]\(\frac{11}{x+7}\)[/tex]
- J. [tex]\(\frac{35+6 x}{7+x}\)[/tex]
- K. [tex]\(\frac{42+5 x}{7 x}\)[/tex]
The correct answer is:
[tex]\[
\boxed{\frac{42 + 5x}{7x}} \quad \text{(Option K)}
\][/tex]
