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1) $3(x - 4) = 2(-2x + 1)$

Solve for $x$:

Expand both sides: $3x - 12 = -4x + 2$

Move all terms involving $x$ to one side and constants to the other: $3x + 4x = 2 + 12$ $7x = 14$

Solve for $x$: $x = 2$

2) $8q + 6 = 4q - 14$

Solve for $q$:

Move all terms involving $q$ to one side and constants to the other: $8q - 4q = -14 - 6$ $4q = -20$

Solve for $q$: $q = -5$

3) $9 = \frac{v + 4}{v + 12}$

Solve for $v$:

Multiply both sides by $v + 12$: $9(v + 12) = v + 4$ $9v + 108 = v + 4$

Move all terms involving $v$ to one side and constants to the other: $9v - v = 4 - 108$ $8v = -104$

Solve for $v$: $v = -13$

4) $7 - (5t - 13) = -25$

Solve for $t$:

Expand and simplify: $7 - 5t + 13 = -25$ $20 - 5t = -25$

Move constants to one side: $-5t = -25 - 20$ $-5t = -45$

Solve for $t$: $t = 9$

5) $-3(7p + 5) = 27$

Solve for $p$:

Expand: $-21p - 15 = 27$

Move constants to one side: $-21p = 27 + 15$ $-21p = 42$

Solve for $p$: $p = -2$

6) $14 + 13y = 20y - 21$

Solve for $y$:

Move all terms involving $y$ to one side and constants to the other: $14 + 21 = 20y - 13y$ $35 = 7y$

Solve for $y$: $y = 5$

7) $\frac{-8 - 3k}{2} = 11$

Solve for $k$:

Multiply both sides by 2: $-8 - 3k = 22$

Move constants to one side: $-3k = 22 + 8$ $-3k = 30$

Solve for $k$: $k = -10$

8) $-15b + 21 + 5b = -19$

Solve for $b$:

Combine like terms: $-10b + 21 = -19$

Move constants to one side: $-10b = -19 - 21$ $-10b = -40$

Solve for $b$: $b = 4$

Would you like any further clarifications on these solutions?

Here are five related questions to deepen your understanding:

1. How do you handle equations with fractions and variables?
2. Can you explain the difference between combining like terms and distributing?
3. What is the process for moving constants to one side of the equation?
4. How does solving for a negative variable change the solution process?
5. How do you verify if the solution to an equation is correct?

Tip: When solving equations, always perform the same operation on both sides to maintain balance!