To solve the equation:

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

Step-by-Step Solution:

Step 1: Eliminate fractions

Find the least common multiple (LCM) of the denominators (5 and 7). The LCM is $35$. Multiply through the entire equation by $35$ to eliminate fractions:

$35 \cdot \left(\frac{1}{5}x + \frac{6}{5}\right) = 35 \cdot \left(\frac{1}{7}x + \frac{8}{7}\right)$

This simplifies to:

$7x + 42 = 5x + 40$

Step 2: Simplify and solve for $x$

Rearrange terms to isolate $x$:

$7x - 5x = 40 - 42$

$2x = -2$

$x = -1$

Final Answer:

$x = -1$

Matching with Options:

The correct choice is B) -1.

Let me know if you'd like further clarification or help with similar problems!

Relative Questions to Practice:

1. How do you solve equations with different denominators?
2. What is the role of the LCM in clearing fractions in an equation?
3. How can you verify the solution to a fraction-based equation?
4. What other methods could be used to solve the equation?
5. How do you deal with equations involving variables on both sides?

Tip: Always eliminate fractions early in the process by