Let’s solve each equation step by step.

1. First equation:
   $\frac{x}{3} = 9$
   Multiply both sides by 3:
   $x = 27$

2. Second equation:
   $\frac{x - 3}{5} = 9$
   Multiply both sides by 5:
   $x - 3 = 45$
   Add 3 to both sides:
   $x = 48$

3. Third equation:
   $\frac{2x - 8}{2} = -1$
   Simplify the left side:
   $x - 4 = -1$
   Add 4 to both sides:
   $x = 3$

4. Fourth equation:
   $\frac{x + 2}{6} = x - 8$
   Multiply both sides by 6:
   $x + 2 = 6(x - 8)$
   Expand:
   $x + 2 = 6x - 48$
   Subtract $x$ from both sides:
   $2 = 5x - 48$
   Add 48 to both sides:
   $50 = 5x$
   Divide by 5:
   $x = 10$

5. Fifth equation:
   $\frac{x - 7}{2x + 4} = 1$
   Multiply both sides by $2x + 4$:
   $x - 7 = 2x + 4$
   Subtract $x$ from both sides:
   $-7 = x + 4$
   Subtract 4 from both sides:
   $x = -11$

6. Sixth equation:
   $\frac{6}{x + 5} = \frac{-3}{x - 7}$
   Cross-multiply:
   $6(x - 7) = -3(x + 5)$
   Expand:
   $6x - 42 = -3x - 15$
   Add $3x$ to both sides:
   $9x - 42 = -15$
   Add 42 to both sides:
   $9x = 27$
   Divide by 9:
   $x = 3$

Final Answers:

1. $x = 27$
2. $x = 48$
3. $x = 3$
4. $x = 10$
5. $x = -11$
6. $x = 3$

Would you like further clarification on any of the steps or additional explanations?

Related Questions:

1. What is the process for solving fractional equations with multiple terms?
2. How can cross-multiplication simplify solving equations with fractions?
3. What are some common mistakes when solving equations involving fractions?
4. How do we check the solutions of equations for accuracy?
5. Why is it important to isolate $x$ when solving equations?

Tip: Always verify