Structured document form displaying multiple data entry sections with highlighted fields for easy identification.

The task involves solving a series of algebraic equations. Below, I will solve each equation step by step and provide the solution.

---

Problem (1):

\[ 4x + 6 = 2 \]

1. Subtract 6 from both sides:
\[
4x + 6 - 6 = 2 - 6
\]
\[
4x = -4
\]

2. Divide both sides by 4:
\[
\frac{4x}{4} = \frac{-4}{4}
\]
\[
x = -1
\]

Solution:
\[
\boxed{x = -1}
\]

---

Problem (2):

\[ 3x - 7 = 8 \]

1. Add 7 to both sides:
\[
3x - 7 + 7 = 8 + 7
\]
\[
3x = 15
\]

2. Divide both sides by 3:
\[
\frac{3x}{3} = \frac{15}{3}
\]
\[
x = 5
\]

Solution:
\[
\boxed{x = 5}
\]

---

Problem (3):

\[ 1 - 2x = -3 \]

1. Subtract 1 from both sides:
\[
1 - 2x - 1 = -3 - 1
\]
\[
-2x = -4
\]

2. Divide both sides by -2:
\[
\frac{-2x}{-2} = \frac{-4}{-2}
\]
\[
x = 2
\]

Solution:
\[
\boxed{x = 2}
\]

---

Problem (4):

\[ 4 - 3(2 - x) = 5 \]

1. Distribute the -3 inside the parentheses:
\[
4 - 3 \cdot 2 + 3 \cdot x = 5
\]
\[
4 - 6 + 3x = 5
\]

2. Simplify the left side:
\[
-2 + 3x = 5
\]

3. Add 2 to both sides:
\[
-2 + 3x + 2 = 5 + 2
\]
\[
3x = 7
\]

4. Divide both sides by 3:
\[
\frac{3x}{3} = \frac{7}{3}
\]
\[
x = \frac{7}{3}
\]

Solution:
\[
\boxed{x = \frac{7}{3}}
\]

---

Problem (5):

\[ 4x - 3(20 - x) = 3 \]

1. Distribute the -3 inside the parentheses:
\[
4x - 3 \cdot 20 + 3 \cdot x = 3
\]
\[
4x - 60 + 3x = 3
\]

2. Combine like terms:
\[
7x - 60 = 3
\]

3. Add 60 to both sides:
\[
7x - 60 + 60 = 3 + 60
\]
\[
7x = 63
\]

4. Divide both sides by 7:
\[
\frac{7x}{7} = \frac{63}{7}
\]
\[
x = 9
\]

Solution:
\[
\boxed{x = 9}
\]

---

Problem (6):

\[ 5(x - 1) = 1 \]

1. Distribute the 5 inside the parentheses:
\[
5 \cdot x - 5 \cdot 1 = 1
\]
\[
5x - 5 = 1
\]

2. Add 5 to both sides:
\[
5x - 5 + 5 = 1 + 5
\]
\[
5x = 6
\]

3. Divide both sides by 5:
\[
\frac{5x}{5} = \frac{6}{5}
\]
\[
x = \frac{6}{5}
\]

Solution:
\[
\boxed{x = \frac{6}{5}}
\]

---

Problem (7):

\[ 3x - 7 + x = 3 - 2x + 3 \]

1. Combine like terms on both sides:
\[
(3x + x) - 7 = 3 + 3 - 2x
\]
\[
4x - 7 = 6 - 2x
\]

2. Add 2x to both sides:
\[
4x + 2x - 7 = 6 - 2x + 2x
\]
\[
6x - 7 = 6
\]

3. Add 7 to both sides:
\[
6x - 7 + 7 = 6 + 7
\]
\[
6x = 13
\]

4. Divide both sides by 6:
\[
\frac{6x}{6} = \frac{13}{6}
\]
\[
x = \frac{13}{6}
\]

Solution:
\[
\boxed{x = \frac{13}{6}}
\]

---

Problem (8):

\[ 5(x - 1) = 3(x + 1) \]

1. Distribute on both sides:
\[
5 \cdot x - 5 \cdot 1 = 3 \cdot x + 3 \cdot 1
\]
\[
5x - 5 = 3x + 3
\]

2. Subtract 3x from both sides:
\[
5x - 3x - 5 = 3x - 3x + 3
\]
\[
2x - 5 = 3
\]

3. Add 5 to both sides:
\[
2x - 5 + 5 = 3 + 5
\]
\[
2x = 8
\]

4. Divide both sides by 2:
\[
\frac{2x}{2} = \frac{8}{2}
\]
\[
x = 4
\]

Solution:
\[
\boxed{x = 4}
\]

---

Problem (9):

\[ 2(x - 2) - 3(4x - 1) = 9(1 - x) \]

1. Distribute on all terms:
\[
2 \cdot x - 2 \cdot 2 - 3 \cdot 4x + 3 \cdot 1 = 9 \cdot 1 - 9 \cdot x
\]
\[
2x - 4 - 12x + 3 = 9 - 9x
\]

2. Combine like terms on the left side:
\[
(2x - 12x) + (-4 + 3) = 9 - 9x
\]
\[
-10x - 1 = 9 - 9x
\]

3. Add 9x to both sides:
\[
-10x + 9x - 1 = 9 - 9x + 9x
\]
\[
-x - 1 = 9
\]

4. Add 1 to both sides:
\[
-x - 1 + 1 = 9 + 1
\]
\[
-x = 10
\]

5. Multiply both sides by -1:
\[
x = -10
\]

Solution:
\[
\boxed{x = -10}
\]

---

Problem (10):

\[ 2(x - 2) - 3(4x - 1) = 9(1 - x) \]

This is the same as Problem (9), so the solution is:
\[
\boxed{x = -10}
\]

---

Final Answers:

\[
\boxed{
\begin{array}{ll}
(1) & x = -1 \\
(2) & x = 5 \\
(3) & x = 2 \\
(4) & x = \frac{7}{3} \\
(5) & x = 9 \\
(6) & x = \frac{6}{5} \\
(7) & x = \frac{13}{6} \\
(8) & x = 4 \\
(9) & x = -10 \\
(10) & x = -10 \\
\end{array}
}
\]