We are given five linear equations to solve. We need to solve for $x$ in equations 1 through 4, and determine the type of solution (no solution, infinitely many solutions, or one solution) for equation 5.

1. Problem Description

We are given five linear equations to solve. We need to solve for $x$ in equations 1 through 4, and determine the type of solution (no solution, infinitely many solutions, or one solution) for equation

5.

2. Solution Steps

Problem 1: $\frac{3}{4}(x+20) = 9$

Step 1: Multiply both sides by 4/3:

$x+20 = 9 \cdot \frac{4}{3}$

Step 2: Simplify the right side:

$x+20 = 12$

Step 3: Subtract 20 from both sides:

$x = 12 - 20$

Step 4: Simplify:

$x = -8$

Problem 2: $4 - 2(x-5) - x = -9$

Step 1: Distribute the -2:

$4 - 2x + 10 - x = -9$

Step 2: Combine like terms:

$14 - 3x = -9$

Step 3: Subtract 14 from both sides:

$-3x = -9 - 14$

Step 4: Simplify:

$-3x = -23$

Step 5: Divide both sides by -3:

$x = \frac{-23}{-3}$

Step 6: Simplify:

$x = \frac{23}{3}$

Problem 3: $3(x-6) - 8x = -2 + 5(2x+1)$

Step 1: Distribute:

$3x - 18 - 8x = -2 + 10x + 5$

Step 2: Combine like terms:

$-5x - 18 = 10x + 3$

Step 3: Add $5x$ to both sides:

$-18 = 15x + 3$

Step 4: Subtract 3 from both sides:

$-21 = 15x$

Step 5: Divide both sides by 15:

$x = \frac{-21}{15}$

Step 6: Simplify the fraction:

$x = -\frac{7}{5}$

Problem 4: $7(2x-3) - \frac{3}{5}x = \frac{6}{5}(10-3x) + 1$

Step 1: Distribute:

$14x - 21 - \frac{3}{5}x = \frac{60}{5} - \frac{18}{5}x + 1$

Step 2: Simplify:

$14x - 21 - \frac{3}{5}x = 12 - \frac{18}{5}x + 1$

Step 3: Combine like terms:

$14x - \frac{3}{5}x + \frac{18}{5}x = 13 + 21$

Step 4: Simplify:

$\frac{70x}{5} - \frac{3x}{5} + \frac{18x}{5} = 34$

Step 5: Combine the $x$ terms:

$\frac{85x}{5} = 34$

Step 6: Simplify:

$17x = 34$

Step 7: Divide both sides by 17:

$x = \frac{34}{17}$

Step 8: Simplify:

$x = 2$

Problem 5: $-2(11-12x) = -4(1-6x)$

Step 1: Distribute:

$-22 + 24x = -4 + 24x$

Step 2: Subtract 24x from both sides:

$-22 = -4$

This statement is false. Therefore, there is no solution.

3. Final Answer

Problem 1: $x = -8$

Problem 2: $x = \frac{23}{3}$

Problem 3: $x = -\frac{7}{5}$

Problem 4: $x = 2$

Problem 5: No solution.