Let's work through each equation step by step.

Equation 1:

$x + 7 = \frac{17}{6} - \frac{5x}{6}$

1. Multiply through by 6 to eliminate fractions: $6(x + 7) = 17 - 5x$ $6x + 42 = 17 - 5x$

2. Add $5x$ to both sides: $11x + 42 = 17$

3. Subtract 42 from both sides: $11x = 17 - 42$ $11x = -25$

4. Divide by 11: $x = -\frac{25}{11}$

Equation 2:

$y - \frac{y - 5}{2} = 1 - 2y$

1. Multiply through by 2 to eliminate fractions: $2y - (y - 5) = 2(1 - 2y)$ $2y - y + 5 = 2 - 4y$

2. Simplify: $y + 5 = 2 - 4y$

3. Add $4y$ to both sides: $5y + 5 = 2$

4. Subtract 5 from both sides: $5y = 2 - 5$ $5y = -3$

5. Divide by 5: $y = -\frac{3}{5}$

Equation 3:

$15(x - 4) - 2(x - 9) = 2(x - 1)$

1. Expand the brackets: $15x - 60 - 2x + 18 = 2x - 2$ $13x - 42 = 2x - 2$

2. Subtract $2x$ from both sides: $11x - 42 = -2$

3. Add 42 to both sides: $11x = 40$

4. Divide by 11: $x = \frac{40}{11}$

Equation 4:

$\frac{m - 2}{3} + 1 = \frac{2m}{7}$

1. Multiply through by 21 to eliminate fractions: $7(m - 2) + 21 = 3(2m)$ $7m - 14 + 21 = 6m$

2. Simplify: $7m + 7 = 6m$

3. Subtract $6m$ from both sides: $m + 7 = 0$

4. Subtract 7 from both sides: $m = -7$

Equation 5:

$\frac{a + 2}{4} + \frac{a - 3}{5} = \frac{3a - 5}{8}$

1. Find the least common denominator (LCD) which is 40, and multiply through: $40 \left( \frac{a + 2}{4} \right) + 40 \left( \frac{a - 3}{5} \right) = 40 \left( \frac{3a - 5}{8} \right)$ $10(a + 2) + 8(a - 3) = 5(3a - 5)$

2. Expand the brackets: $10a + 20 + 8a - 24 = 15a - 25$ $18a - 4 = 15a - 25$

3. Subtract $15a$ from both sides: $3a - 4 = -25$

4. Add 4 to both sides: $3a = -21$

5. Divide by 3: $a = -7$

Equation 6:

$\frac{3x - 1}{4} - \frac{2x + 2}{6} = \frac{5}{2} - 2x$

1. Find the least common denominator, which is 12, and multiply through: $12 \left( \frac{3x - 1}{4} \right) - 12 \left( \frac{2x + 2}{6} \right) = 12 \left( \frac{5}{2} \right) - 12(2x)$ $3(3x - 1) - 2(2x + 2) = 6(5) - 24x$

2. Expand the brackets: $9x - 3 - 4x - 4 = 30 - 24x$ $5x - 7 = 30 - 24x$

3. Add $24x$ to both sides: $29x - 7 = 30$

4. Add 7 to both sides: $29x = 37$

5. Divide by 29: $x = \frac{37}{29}$

Would you like further details on any of these solutions or verification steps?

Here are 5 related questions for practice:

1. How can you solve equations with more complex fractions?
2. What is the best approach to handle equations with variables on both sides?
3. How do you verify the solution of an equation after solving it?
4. How does clearing fractions help simplify equations?
5. Can you solve quadratic equations using a similar approach?

Tip: Always double-check for common denominators when dealing with fractions to simplify calculations!