Linear Equations Worksheet for solving algebraic equations.

Here's the step-by-step solution to each of the 8 linear equations on the worksheet.

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① 8(4a - 4) - 1 = 14 - 8a

Step 1: Distribute the 8 on the left side:
→ 32a - 32 - 1 = 14 - 8a
→ 32a - 33 = 14 - 8a

Step 2: Add 8a to both sides:
→ 32a + 8a - 33 = 14
→ 40a - 33 = 14

Step 3: Add 33 to both sides:
→ 40a = 47

Step 4: Divide by 40:
→ a = 47/40

✔ Answer: a = 47/40

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② 4(-8x + 5) = -30x - 26

Step 1: Distribute the 4:
→ -32x + 20 = -30x - 26

Step 2: Add 32x to both sides:
→ 20 = 2x - 26

Step 3: Add 26 to both sides:
→ 46 = 2x

Step 4: Divide by 2:
→ x = 23

✔ Answer: x = 23

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③ 27 = 46 + 3x - x

Step 1: Combine like terms on the right:
→ 27 = 46 + 2x

Step 2: Subtract 46 from both sides:
→ 27 - 46 = 2x
→ -19 = 2x

Step 3: Divide by 2:
→ x = -19/2

✔ Answer: x = -19/2

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④ 38 + 7k = 8(k + 4)

Step 1: Distribute the 8:
→ 38 + 7k = 8k + 32

Step 2: Subtract 7k from both sides:
→ 38 = k + 32

Step 3: Subtract 32 from both sides:
→ k = 6

✔ Answer: k = 6

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⑤ -2(4 - x) = 6(x + 4) + 4x

Step 1: Distribute both sides:
Left: -8 + 2x
Right: 6x + 24 + 4x = 10x + 24
→ -8 + 2x = 10x + 24

Step 2: Subtract 2x from both sides:
→ -8 = 8x + 24

Step 3: Subtract 24 from both sides:
→ -32 = 8x

Step 4: Divide by 8:
→ x = -4

✔ Answer: x = -4

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⑥ 0.05z + 0.2 = 0.15z - 10.5

Step 1: Subtract 0.05z from both sides:
→ 0.2 = 0.10z - 10.5

Step 2: Add 10.5 to both sides:
→ 10.7 = 0.10z

Step 3: Divide by 0.10 (or multiply by 10):
→ z = 107

✔ Answer: z = 107

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⑦ 4(6 + 4x) = 0

Step 1: Distribute the 4:
→ 24 + 16x = 0

Step 2: Subtract 24 from both sides:
→ 16x = -24

Step 3: Divide by 16:
→ x = -24/16 = -3/2

✔ Answer: x = -3/2

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⑧ x/8 - x/9 = 1

Step 1: Find a common denominator for the fractions (LCM of 8 and 9 is 72):

→ (9x)/72 - (8x)/72 = 1
→ (9x - 8x)/72 = 1
→ x/72 = 1

Step 2: Multiply both sides by 72:
→ x = 72

✔ Answer: x = 72

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## ✔ Final Answers:

1. a = 47/40
2. x = 23
3. x = -19/2
4. k = 6
5. x = -4
6. z = 107
7. x = -3/2
8. x = 72

Let me know if you’d like to see any step explained in more detail!