This worksheet provides a comprehensive set of linear equation problems, progressing from simple forms to those involving fractions and brackets.

Let's solve each equation step-by-step, explaining the process clearly. We'll go through Section A, Section B, and Section C in order.

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✔ Section A: Basic Linear Equations

These equations are straightforward and involve basic operations.

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#### 1) $ 4x + 10 = 30 $

Subtract 10 from both sides:
$ 4x = 20 $
Divide by 4:
$ x = 5 $

✔ Answer: 5

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#### 2) $ 4x - 8 = 20 $

Add 8 to both sides:
$ 4x = 28 $
Divide by 4:
$ x = 7 $

✔ Answer: 7

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#### 3) $ 5 + 2x = 65 $

Subtract 5:
$ 2x = 60 $
Divide by 2:
$ x = 30 $

✔ Answer: 30

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#### 4) $ 9 + 4x = -15 $

Subtract 9:
$ 4x = -24 $
Divide by 4:
$ x = -6 $

✔ Answer: -6

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#### 5) $ 14 + 6x = 2 $

Subtract 14:
$ 6x = -12 $
Divide by 6:
$ x = -2 $

✔ Answer: -2

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#### 6) $ 2x - 3 = -2 $

Add 3:
$ 2x = 1 $
Divide by 2:
$ x = \frac{1}{2} $

✔ Answer: $ \frac{1}{2} $

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#### 7) $ 5 + 10x = -15 $

Subtract 5:
$ 10x = -20 $
Divide by 10:
$ x = -2 $

✔ Answer: -2

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#### 8) $ 10 = 7 - x $

Rewrite: $ 10 = 7 - x $
Subtract 7:
$ 3 = -x $
Multiply both sides by -1:
$ x = -3 $

✔ Answer: -3

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#### 9) $ -3 = 16 - x $

Subtract 16:
$ -19 = -x $
Multiply by -1:
$ x = 19 $

✔ Answer: 19

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#### 10) $ -4 = 12 - 2x $

Subtract 12:
$ -16 = -2x $
Divide by -2:
$ x = 8 $

✔ Answer: 8

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#### 11) $ 25 = 46 - 3x $

Subtract 46:
$ -21 = -3x $
Divide by -3:
$ x = 7 $

✔ Answer: 7

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#### 12) $ 8 = 9 - 5x $

Subtract 9:
$ -1 = -5x $
Divide by -5:
$ x = \frac{1}{5} $

✔ Answer: $ \frac{1}{5} $

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✔ Section B: Equations with Fractions

We’ll isolate $ x $ and eliminate fractions by multiplying.

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#### 1) $ \frac{x}{2} + 11 = 19 $

Subtract 11:
$ \frac{x}{2} = 8 $
Multiply by 2:
$ x = 16 $

✔ Answer: 16

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#### 2) $ \frac{x}{7} - 6 = 1 $

Add 6:
$ \frac{x}{7} = 7 $
Multiply by 7:
$ x = 49 $

✔ Answer: 49

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#### 3) $ 12 + \frac{x}{5} = 20 $

Subtract 12:
$ \frac{x}{5} = 8 $
Multiply by 5:
$ x = 40 $

✔ Answer: 40

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#### 4) $ 3 = \frac{x}{4} - 3 $

Add 3:
$ 6 = \frac{x}{4} $
Multiply by 4:
$ x = 24 $

✔ Answer: 24

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#### 5) $ 7 = \frac{x}{2} - 4 $

Add 4:
$ 11 = \frac{x}{2} $
Multiply by 2:
$ x = 22 $

✔ Answer: 22

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#### 6) $ -2 = \frac{x}{8} - 5 $

Add 5:
$ 3 = \frac{x}{8} $
Multiply by 8:
$ x = 24 $

✔ Answer: 24

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#### 7) $ -1 = 6 + \frac{x}{2} $

Subtract 6:
$ -7 = \frac{x}{2} $
Multiply by 2:
$ x = -14 $

✔ Answer: -14

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#### 8) $ 14 - \frac{x}{3} = 10 $

Subtract 14:
$ -\frac{x}{3} = -4 $
Multiply by -3:
$ x = 12 $

✔ Answer: 12

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#### 9) $ 5 - \frac{x}{9} = -1 $

Subtract 5:
$ -\frac{x}{9} = -6 $
Multiply by -9:
$ x = 54 $

✔ Answer: 54

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#### 10) $ \frac{x + 5}{3} = 12 $

Multiply both sides by 3:
$ x + 5 = 36 $
Subtract 5:
$ x = 31 $

✔ Answer: 31

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#### 11) $ \frac{x - 4}{11} = 9 $

Multiply by 11:
$ x - 4 = 99 $
Add 4:
$ x = 103 $

✔ Answer: 103

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#### 12) $ \frac{x + 3}{8} = -2 $

Multiply by 8:
$ x + 3 = -16 $
Subtract 3:
$ x = -19 $

✔ Answer: -19

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✔ Section C: Equations with Brackets

We’ll expand brackets first, then simplify.

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#### 1) $ 3(x + 2) = 15 $

Expand:
$ 3x + 6 = 15 $
Subtract 6:
$ 3x = 9 $
Divide by 3:
$ x = 3 $

✔ Answer: 3

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#### 2) $ 2(x + 5) = 24 $

Expand:
$ 2x + 10 = 24 $
Subtract 10:
$ 2x = 14 $
Divide by 2:
$ x = 7 $

✔ Answer: 7

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#### 3) $ 6(x - 9) = 12 $

Expand:
$ 6x - 54 = 12 $
Add 54:
$ 6x = 66 $
Divide by 6:
$ x = 11 $

✔ Answer: 11

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#### 4) $ 2(3x + 5) = -44 $

Expand:
$ 6x + 10 = -44 $
Subtract 10:
$ 6x = -54 $
Divide by 6:
$ x = -9 $

✔ Answer: -9

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#### 5) $ 5(4x - 3) = 11 $

Expand:
$ 20x - 15 = 11 $
Add 15:
$ 20x = 26 $
Divide by 20:
$ x = \frac{26}{20} = \frac{13}{10} = 1.3 $

✔ Answer: $ \frac{13}{10} $ or 1.3

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#### 6) $ -3(2x + 1) = 21 $

Expand:
$ -6x - 3 = 21 $
Add 3:
$ -6x = 24 $
Divide by -6:
$ x = -4 $

✔ Answer: -4

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#### 7) $ -9(x - 4) = 54 $

Expand:
$ -9x + 36 = 54 $
Subtract 36:
$ -9x = 18 $
Divide by -9:
$ x = -2 $

✔ Answer: -2

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#### 8) $ 7(x - 4) - 3 = 46 $

Expand:
$ 7x - 28 - 3 = 46 $ → $ 7x - 31 = 46 $
Add 31:
$ 7x = 77 $
Divide by 7:
$ x = 11 $

✔ Answer: 11

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#### 9) $ 2(3x - 1) + 3 = 21 $

Expand:
$ 6x - 2 + 3 = 21 $ → $ 6x + 1 = 21 $
Subtract 1:
$ 6x = 20 $
Divide by 6:
$ x = \frac{20}{6} = \frac{10}{3} \approx 3.33 $

✔ Answer: $ \frac{10}{3} $

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#### 10) $ 2(x + 1) + 3x = 37 $

Expand:
$ 2x + 2 + 3x = 37 $ → $ 5x + 2 = 37 $
Subtract 2:
$ 5x = 35 $
Divide by 5:
$ x = 7 $

✔ Answer: 7

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#### 11) $ 12 + 4(2x + 4) = 68 $

Expand:
$ 12 + 8x + 16 = 68 $ → $ 8x + 28 = 68 $
Subtract 28:
$ 8x = 40 $
Divide by 8:
$ x = 5 $

✔ Answer: 5

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#### 12) $ 3x - 2(6x - 3) = 42 $

Expand:
$ 3x - 12x + 6 = 42 $ → $ -9x + 6 = 42 $
Subtract 6:
$ -9x = 36 $
Divide by -9:
$ x = -4 $

✔ Answer: -4

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✔ Final Answers Summary

#### Section A
1. 5
2. 7
3. 30
4. -6
5. -2
6. $ \frac{1}{2} $
7. -2
8. -3
9. 19
10. 8
11. 7
12. $ \frac{1}{5} $

#### Section B
1. 16
2. 49
3. 40
4. 24
5. 22
6. 24
7. -14
8. 12
9. 54
10. 31
11. 103
12. -19

#### Section C
1. 3
2. 7
3. 11
4. -9
5. $ \frac{13}{10} $ (or 1.3)
6. -4
7. -2
8. 11
9. $ \frac{10}{3} $
10. 7
11. 5
12. -4

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