Solving Linear Equations worksheet for Grade D students, including problems with negative, fractional, and decimal solutions.

Problem: Solve the given linear equations.

The task involves solving a series of linear equations, some of which may have negative, fractional, or decimal solutions. Below, I will solve each equation step by step.

---

Section A

#### 1. \( 4x + 10 = 30 \)
1. Subtract 10 from both sides:
\[
4x + 10 - 10 = 30 - 10
\]
\[
4x = 20
\]
2. Divide both sides by 4:
\[
x = \frac{20}{4}
\]
\[
x = 5
\]

Answer: \( x = 5 \)

---

#### 2. \( 4x - 8 = 20 \)
1. Add 8 to both sides:
\[
4x - 8 + 8 = 20 + 8
\]
\[
4x = 28
\]
2. Divide both sides by 4:
\[
x = \frac{28}{4}
\]
\[
x = 7
\]

Answer: \( x = 7 \)

---

#### 3. \( 5 + 2x = 65 \)
1. Subtract 5 from both sides:
\[
5 + 2x - 5 = 65 - 5
\]
\[
2x = 60
\]
2. Divide both sides by 2:
\[
x = \frac{60}{2}
\]
\[
x = 30
\]

Answer: \( x = 30 \)

---

#### 4. \( 9 + 4x = -15 \)
1. Subtract 9 from both sides:
\[
9 + 4x - 9 = -15 - 9
\]
\[
4x = -24
\]
2. Divide both sides by 4:
\[
x = \frac{-24}{4}
\]
\[
x = -6
\]

Answer: \( x = -6 \)

---

#### 5. \( 14 + 6x = 2 \)
1. Subtract 14 from both sides:
\[
14 + 6x - 14 = 2 - 14
\]
\[
6x = -12
\]
2. Divide both sides by 6:
\[
x = \frac{-12}{6}
\]
\[
x = -2
\]

Answer: \( x = -2 \)

---

#### 6. \( 2x - 3 = -2 \)
1. Add 3 to both sides:
\[
2x - 3 + 3 = -2 + 3
\]
\[
2x = 1
\]
2. Divide both sides by 2:
\[
x = \frac{1}{2}
\]

Answer: \( x = \frac{1}{2} \)

---

#### 7. \( 5 + 10x = -15 \)
1. Subtract 5 from both sides:
\[
5 + 10x - 5 = -15 - 5
\]
\[
10x = -20
\]
2. Divide both sides by 10:
\[
x = \frac{-20}{10}
\]
\[
x = -2
\]

Answer: \( x = -2 \)

---

#### 8. \( 10 = 7 - x \)
1. Subtract 7 from both sides:
\[
10 - 7 = 7 - x - 7
\]
\[
3 = -x
\]
2. Multiply both sides by -1:
\[
x = -3
\]

Answer: \( x = -3 \)

---

#### 9. \( -3 = 16 - x \)
1. Subtract 16 from both sides:
\[
-3 - 16 = 16 - x - 16
\]
\[
-19 = -x
\]
2. Multiply both sides by -1:
\[
x = 19
\]

Answer: \( x = 19 \)

---

#### 10. \( -4 = 12 - 2x \)
1. Subtract 12 from both sides:
\[
-4 - 12 = 12 - 2x - 12
\]
\[
-16 = -2x
\]
2. Divide both sides by -2:
\[
x = \frac{-16}{-2}
\]
\[
x = 8
\]

Answer: \( x = 8 \)

---

#### 11. \( 25 = 46 - 3x \)
1. Subtract 46 from both sides:
\[
25 - 46 = 46 - 3x - 46
\]
\[
-21 = -3x
\]
2. Divide both sides by -3:
\[
x = \frac{-21}{-3}
\]
\[
x = 7
\]

Answer: \( x = 7 \)

---

#### 12. \( 8 = 9 - 5x \)
1. Subtract 9 from both sides:
\[
8 - 9 = 9 - 5x - 9
\]
\[
-1 = -5x
\]
2. Divide both sides by -5:
\[
x = \frac{-1}{-5}
\]
\[
x = \frac{1}{5}
\]

Answer: \( x = \frac{1}{5} \)

---

Section B

#### 1. \( \frac{x}{2} + 11 = 19 \)
1. Subtract 11 from both sides:
\[
\frac{x}{2} + 11 - 11 = 19 - 11
\]
\[
\frac{x}{2} = 8
\]
2. Multiply both sides by 2:
\[
x = 8 \cdot 2
\]
\[
x = 16
\]

Answer: \( x = 16 \)

---

#### 2. \( \frac{x}{7} - 6 = 1 \)
1. Add 6 to both sides:
\[
\frac{x}{7} - 6 + 6 = 1 + 6
\]
\[
\frac{x}{7} = 7
\]
2. Multiply both sides by 7:
\[
x = 7 \cdot 7
\]
\[
x = 49
\]

Answer: \( x = 49 \)

---

#### 3. \( 12 + \frac{x}{5} = 20 \)
1. Subtract 12 from both sides:
\[
12 + \frac{x}{5} - 12 = 20 - 12
\]
\[
\frac{x}{5} = 8
\]
2. Multiply both sides by 5:
\[
x = 8 \cdot 5
\]
\[
x = 40
\]

Answer: \( x = 40 \)

---

#### 4. \( 3 = \frac{x}{4} - 3 \)
1. Add 3 to both sides:
\[
3 + 3 = \frac{x}{4} - 3 + 3
\]
\[
6 = \frac{x}{4}
\]
2. Multiply both sides by 4:
\[
x = 6 \cdot 4
\]
\[
x = 24
\]

Answer: \( x = 24 \)

---

#### 5. \( 7 = \frac{x}{2} - 4 \)
1. Add 4 to both sides:
\[
7 + 4 = \frac{x}{2} - 4 + 4
\]
\[
11 = \frac{x}{2}
\]
2. Multiply both sides by 2:
\[
x = 11 \cdot 2
\]
\[
x = 22
\]

Answer: \( x = 22 \)

---

#### 6. \( -2 = \frac{x}{8} - 5 \)
1. Add 5 to both sides:
\[
-2 + 5 = \frac{x}{8} - 5 + 5
\]
\[
3 = \frac{x}{8}
\]
2. Multiply both sides by 8:
\[
x = 3 \cdot 8
\]
\[
x = 24
\]

Answer: \( x = 24 \)

---

#### 7. \( -1 = 6 + \frac{x}{2} \)
1. Subtract 6 from both sides:
\[
-1 - 6 = 6 + \frac{x}{2} - 6
\]
\[
-7 = \frac{x}{2}
\]
2. Multiply both sides by 2:
\[
x = -7 \cdot 2
\]
\[
x = -14
\]

Answer: \( x = -14 \)

---

#### 8. \( 14 - \frac{x}{3} = 10 \)
1. Subtract 14 from both sides:
\[
14 - \frac{x}{3} - 14 = 10 - 14
\]
\[
-\frac{x}{3} = -4
\]
2. Multiply both sides by -3:
\[
x = (-4) \cdot (-3)
\]
\[
x = 12
\]

Answer: \( x = 12 \)

---

#### 9. \( 5 - \frac{x}{9} = -1 \)
1. Subtract 5 from both sides:
\[
5 - \frac{x}{9} - 5 = -1 - 5
\]
\[
-\frac{x}{9} = -6
\]
2. Multiply both sides by -9:
\[
x = (-6) \cdot (-9)
\]
\[
x = 54
\]

Answer: \( x = 54 \)

---

#### 10. \( \frac{x + 5}{3} = 12 \)
1. Multiply both sides by 3:
\[
x + 5 = 12 \cdot 3
\]
\[
x + 5 = 36
\]
2. Subtract 5 from both sides:
\[
x = 36 - 5
\]
\[
x = 31
\]

Answer: \( x = 31 \)

---

#### 11. \( \frac{x - 4}{11} = 9 \)
1. Multiply both sides by 11:
\[
x - 4 = 9 \cdot 11
\]
\[
x - 4 = 99
\]
2. Add 4 to both sides:
\[
x = 99 + 4
\]
\[
x = 103
\]

Answer: \( x = 103 \)

---

#### 12. \( \frac{x + 3}{8} = -2 \)
1. Multiply both sides by 8:
\[
x + 3 = -2 \cdot 8
\]
\[
x + 3 = -16
\]
2. Subtract 3 from both sides:
\[
x = -16 - 3
\]
\[
x = -19
\]

Answer: \( x = -19 \)

---

Final Answers:

Section A:
1. \( x = 5 \)
2. \( x = 7 \)
3. \( x = 30 \)
4. \( x = -6 \)
5. \( x = -2 \)
6. \( x = \frac{1}{2} \)
7. \( x = -2 \)
8. \( x = -3 \)
9. \( x = 19 \)
10. \( x = 8 \)
11. \( x = 7 \)
12. \( x = \frac{1}{5} \)

Section B:
1. \( x = 16 \)
2. \( x = 49 \)
3. \( x = 40 \)
4. \( x = 24 \)
5. \( x = 22 \)
6. \( x = 24 \)
7. \( x = -14 \)
8. \( x = 12 \)
9. \( x = 54 \)
10. \( x = 31 \)
11. \( x = 103 \)
12. \( x = -19 \)

\boxed{\text{See detailed solutions above.}}