Here is the solution to your equation:

We are solving: $\frac{1}{5}x + \frac{6}{5} = \frac{1}{7}x + \frac{8}{7}$

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

• The denominators are $5$ and $7$, and their LCD is $35$.
• Multiply the entire equation by $35$:

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

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

Step 2: Simplify and solve for $x$.

• Bring $x$-terms to one side: $7x - 5x = 40 - 42$ $2x = -2$

• Divide by $2$: $x = -1$

Final Answer:

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

Five Similar Problems:

1. Solve: $\frac{1}{3}x + \frac{5}{3} = \frac{2}{4}x + \frac{3}{2}$
2. Solve: $\frac{2}{5}x + \frac{7}{10} = \frac{1}{6}x + \frac{5}{3}$
3. Solve: $\frac{3}{4}x + \frac{4}{5} = \frac{2}{3}x + \frac{7}{6}$
4. Solve: $\frac{5}{6}x + \frac{2}{3} = \frac{4}{9}x + \frac{1}{2}$
5. Solve: $\frac{7}{8}x + \frac{1}{4} = \frac{5}{6}x + \frac{3}{5}$

Tip:

When solving equations with fractions, always eliminate the fractions early by multiplying through by the least common denominator (LCD). This simplifies the equation and reduces the chance of errors. Let me know if you'd like step-by-step solutions for the similar problems or have more questions!