Here are the solutions for the equations:

1. $x + 7 = \frac{17}{6} - \frac{5x}{6}$
   Solution: $x = -2.27$

2. $y - \frac{y - 5}{2} = 1 - 2y$
   Solution: $y = -\frac{3}{5}$

3. $15(x - 4) - 2(x - 9) = 2(x - 1)$
   Solution: $x = \frac{40}{11} \approx 3.64$

4. $\frac{m - 2}{3} + 1 = \frac{2m}{7}$
   Solution: $m = -7$

5. $\frac{a + 2}{4} + \frac{a - 3}{5} = \frac{3a - 5}{8}$
   Solution: $a = -7$

6. $\frac{3x - 1}{4} - \frac{2x + 2}{6} = \frac{5}{2} - 2x$
   Solution: $x \approx 1.28$

Would you like to verify the solutions or ask further questions?

Here are 5 related questions:

1. How do we verify the solutions by substituting them back into the original equations?
2. What are the steps to simplify equations with fractions?
3. How do we handle equations with multiple terms involving variables?
4. Can these equations be solved graphically, and how?
5. How do we solve for one variable when multiple variables are present in an equation?

Tip: Always check your solutions by substituting them back into the original equation to avoid calculation errors.