Year 9 Maths Worksheets | Printable Maths worksheets - Free Printable

Problem: Solving Linear Equations

The task is to solve the given linear equations in three sections. Let's 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:
\[
x = 8 \times 2
\]
\[
x = 16
\]

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:
\[
x = 7 \times 7
\]
\[
x = 49
\]

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:
\[
x = 8 \times 5
\]
\[
x = 40
\]

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:
\[
x = 6 \times 4
\]
\[
x = 24
\]

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:
\[
x = 11 \times 2
\]
\[
x = 22
\]

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:
\[
x = 3 \times 8
\]
\[
x = 24
\]

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:
\[
x = -7 \times 2
\]
\[
x = -14
\]

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:
\[
x = -4 \times -3
\]
\[
x = 12
\]

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:
\[
x = -6 \times -9
\]
\[
x = 54
\]

Answer: \( x = 54 \)

---

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

Answer: \( x = 31 \)

---

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

Answer: \( x = 103 \)

---

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

Answer: \( x = -19 \)

---

Section C

#### 1) \( 3(x + 2) = 15 \)
1. Distribute the 3:
\[
3x + 6 = 15
\]
2. Subtract 6 from both sides:
\[
3x + 6 - 6 = 15 - 6
\]
\[
3x = 9
\]
3. Divide both sides by 3:
\[
x = \frac{9}{3}
\]
\[
x = 3
\]

Answer: \( x = 3 \)

---

#### 2) \( 2(x + 5) = 24 \)
1. Distribute the 2:
\[
2x + 10 = 24
\]
2. Subtract 10 from both sides:
\[
2x + 10 - 10 = 24 - 10
\]
\[
2x = 14
\]
3. Divide both sides by 2:
\[
x = \frac{14}{2}
\]
\[
x = 7
\]

Answer: \( x = 7 \)

---

#### 3) \( 6(x - 9) = 12 \)
1. Distribute the 6:
\[
6x - 54 = 12
\]
2. Add 54 to both sides:
\[
6x - 54 + 54 = 12 + 54
\]
\[
6x = 66
\]
3. Divide both sides by 6:
\[
x = \frac{66}{6}
\]
\[
x = 11
\]

Answer: \( x = 11 \)

---

#### 4) \( 2(3x + 5) = -44 \)
1. Distribute the 2:
\[
6x + 10 = -44
\]
2. Subtract 10 from both sides:
\[
6x + 10 - 10 = -44 - 10
\]
\[
6x = -54
\]
3. Divide both sides by 6:
\[
x = \frac{-54}{6}
\]
\[
x = -9
\]

Answer: \( x = -9 \)

---

#### 5) \( 5(4x - 3) = 11 \)
1. Distribute the 5:
\[
20x - 15 = 11
\]
2. Add 15 to both sides:
\[
20x - 15 + 15 = 11 + 15
\]
\[
20x = 26
\]
3. Divide both sides by 20:
\[
x = \frac{26}{20}
\]
\[
x = \frac{13}{10}
\]

Answer: \( x = \frac{13}{10} \)

---

#### 6) \( -3(2x + 1) = 21 \)
1. Distribute the -3:
\[
-6x - 3 = 21
\]
2. Add 3 to both sides:
\[
-6x - 3 + 3 = 21 + 3
\]
\[
-6x = 24
\]
3. Divide both sides by -6:
\[
x = \frac{24}{-6}
\]
\[
x = -4
\]

Answer: \( x = -4 \)

---

#### 7) \( -9(x - 4) = 54 \)
1. Distribute the -9:
\[
-9x + 36 = 54
\]
2. Subtract 36 from both sides:
\[
-9x + 36 - 36 = 54 - 36
\]
\[
-9x = 18
\]
3. Divide both sides by -9:
\[
x = \frac{18}{-9}
\]
\[
x = -2
\]

Answer: \( x = -2 \)

---

#### 8) \( 7(x - 4) - 3 = 46 \)
1. Distribute the 7:
\[
7x - 28 - 3 = 46
\]
2. Combine like terms:
\[
7x - 31 = 46
\]
3. Add 31 to both sides:
\[
7x - 31 + 31 = 46 + 31
\]
\[
7x = 77
\]
4. Divide both sides by 7:
\[
x = \frac{77}{7}
\]
\[
x = 11
\]

Answer: \( x = 11 \)

---

#### 9) \( 2(3x - 1) + 3 = 21 \)
1. Distribute the 2:
\[
6x - 2 + 3 = 21
\]
2. Combine like terms:
\[
6x + 1 = 21
\]
3. Subtract 1 from both sides:
\[
6x + 1 - 1 = 21 - 1
\]
\[
6x = 20
\]
4. Divide both sides by 6:
\[
x = \frac{20}{6}
\]
\[
x = \frac{10}{3}
\]

Answer: \( x = \frac{10}{3} \)

---

#### 10) \( 2(x + 1) + 3x = 37 \)
1. Distribute the 2:
\[
2x + 2 + 3x = 37
\]
2. Combine like terms:
\[
5x + 2 = 37
\]
3. Subtract 2 from both sides:
\[
5x + 2 - 2 = 37 - 2
\]
\[
5x = 35
\]
4. Divide both sides by 5:
\[
x = \frac{35}{5}
\]
\[
x = 7
\]

Answer: \( x = 7 \)

---

#### 11) \( 12 + 4(2x + 4) = 68 \)
1. Distribute the 4:
\[
12 + 8x + 16 = 68
\]
2. Combine like terms:
\[
8x + 28 = 68
\]
3. Subtract 28 from both sides:
\[
8x + 28 - 28 = 68 - 28
\]
\[
8x = 40
\]
4. Divide both sides by 8:
\[
x = \frac{40}{8}
\]
\[
x = 5
\]

Answer: \( x = 5 \)

---

#### 12) \( 3x - 2(6x - 3) = 42 \)
1. Distribute the -2:
\[
3x - 12x + 6 = 42
\]
2. Combine like terms:
\[
-9x + 6 = 42
\]
3. Subtract 6 from both sides:
\[
-9x + 6 - 6 = 42 - 6
\]
\[
-9x = 36
\]
4. Divide both sides by -9:
\[
x = \frac{36}{-9}
\]
\[
x = -4
\]

Answer: \( x = -4 \)

---

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. \( x = -19 \)

Section C:
1. \( x = 3 \)
2. \( x = 7 \)
3. \( x = 11 \)
4. \( x = -9 \)
5. \( x = \frac{13}{10} \)
6. \( x = -4 \)
7. \( x = -2 \)
8. \( x = 11 \)
9. \( x = \frac{10}{3} \)
10. \( x = 7 \)
11. \( x = 5 \)
12. \( x = -4 \)

Boxed Final Answer:
\[
\boxed{
\begin{array}{lll}
\text{Section A:} & 5, 7, 30, -6, -2, \frac{1}{2}, -2, -3, 19, 8, 7, \frac{1}{5} \\
\text{Section B:} & 16, 49, 40, 24, 22, 24, -14, 12, 54, 31, 103, -19 \\
\text{Section C:} & 3, 7, 11, -9, \frac{13}{10}, -4, -2, 11, \frac{10}{3}, 7, 5, -4 \\
\end{array}
}
\]