Let's solve each problem step-by-step.

Problem 1

$5x = 35$ To solve for $x$, divide both sides by 5: $x = \frac{35}{5} = 7$ Answer: $x = 7$

Problem 2

$24 = -3x$ To solve for $x$, divide both sides by -3: $x = \frac{24}{-3} = -8$ Answer: $x = -8$

Problem 3

$14c + 15 = 43 + 7c$

1. Move $7c$ to the left side by subtracting $7c$ from both sides: $14c - 7c + 15 = 43$ $7c + 15 = 43$
2. Subtract 15 from both sides: $7c = 28$
3. Divide by 7: $c = 4$ Answer: $c = 4$

Problem 4

$5(2y - 1) = 2(4y - 3)$

1. Expand both sides: $10y - 5 = 8y - 6$
2. Move $8y$ to the left side by subtracting $8y$ from both sides: $10y - 8y - 5 = -6$ $2y - 5 = -6$
3. Add 5 to both sides: $2y = -1$
4. Divide by 2: $y = -\frac{1}{2}$ Answer: $y = -\frac{1}{2}$

Problem 5

$5p + 6(p + 7) = 3(p + 2)$

1. Expand both sides: $5p + 6p + 42 = 3p + 6$ $11p + 42 = 3p + 6$
2. Move $3p$ to the left side by subtracting $3p$ from both sides: $11p - 3p + 42 = 6$ $8p + 42 = 6$
3. Subtract 42 from both sides: $8p = -36$
4. Divide by 8: $p = -\frac{36}{8} = -\frac{9}{2}$ Answer: $p = -\frac{9}{2}$

Problem 6

[ 32 - [4 + 6x - 5(x + 4)] = 4(3x + 4) - [6(3x - 4) + 7 - 4x] ]

1. Start by expanding within the brackets on both sides.

For the left side: $32 - [4 + 6x - 5x - 20]$ $= 32 - [ -16 + x]$ [