To solve the equation step by step:

The given equation is: $\frac{1}{5}x + \frac{6}{5} = \frac{1}{7}x + \frac{8}{7}.$

Step 1: Eliminate the fractions by multiplying through by the least common denominator (LCD).

The denominators are $5$ and $7$, so the LCD is $35$. Multiply every term by $35$:

$35 \cdot \frac{1}{5}x + 35 \cdot \frac{6}{5} = 35 \cdot \frac{1}{7}x + 35 \cdot \frac{8}{7}.$

This simplifies to: $7x + 42 = 5x + 40.$

Step 2: Simplify and solve for $x$.

Combine like terms: $7x - 5x = 40 - 42,$ $2x = -2.$

Divide both sides by $2$: $x = -1.$

Final Answer:

The solution is: $\boxed{-1}.$

This corresponds to option $B$.

Let me know if you need further details or explanations!

Relative Questions to Expand Understanding:

1. What is the general method for solving equations with fractions?
2. How do you find the least common denominator (LCD) in a set of fractions?
3. Why does multiplying through by the LCD eliminate fractions?
4. What alternative methods could you use to solve such equations without finding the LCD?
5. How can you verify your solution is correct?

Tip:

When solving equations with fractions, always consider clearing the fractions first to make the process simpler and reduce calculation errors.