Solve the following linear equations: a set of algebra problems for practice.

Let's solve each of the linear equations step by step. The goal is to isolate $ x $ on one side of the equation.

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(a) $ 4x + 1 = 2x + 7 $

Step 1: Subtract $ 2x $ from both sides:
$ 4x - 2x + 1 = 7 $ → $ 2x + 1 = 7 $

Step 2: Subtract 1 from both sides:
$ 2x = 6 $

Step 3: Divide by 2:
$ x = 3 $

✔ Answer: $ x = 3 $

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(b) $ 5x + 4 = 3x + 16 $

Step 1: Subtract $ 3x $:
$ 2x + 4 = 16 $

Step 2: Subtract 4:
$ 2x = 12 $

Step 3: Divide by 2:
$ x = 6 $

✔ Answer: $ x = 6 $

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(c) $ 2x + 8 = x + 12 $

Step 1: Subtract $ x $:
$ x + 8 = 12 $

Step 2: Subtract 8:
$ x = 4 $

✔ Answer: $ x = 4 $

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(d) $ 7x + 1 = 2x + 46 $

Step 1: Subtract $ 2x $:
$ 5x + 1 = 46 $

Step 2: Subtract 1:
$ 5x = 45 $

Step 3: Divide by 5:
$ x = 9 $

✔ Answer: $ x = 9 $

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(e) $ 6x - 3 = 2x + 13 $

Step 1: Subtract $ 2x $:
$ 4x - 3 = 13 $

Step 2: Add 3:
$ 4x = 16 $

Step 3: Divide by 4:
$ x = 4 $

✔ Answer: $ x = 4 $

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(f) $ 9x - 10 = 7x + 24 $

Step 1: Subtract $ 7x $:
$ 2x - 10 = 24 $

Step 2: Add 10:
$ 2x = 34 $

Step 3: Divide by 2:
$ x = 17 $

✔ Answer: $ x = 17 $

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(g) $ 2x + 21 = 4x + 5 $

Step 1: Subtract $ 2x $:
$ 21 = 2x + 5 $

Step 2: Subtract 5:
$ 16 = 2x $

Step 3: Divide by 2:
$ x = 8 $

✔ Answer: $ x = 8 $

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(h) $ x + 2 = 5x - 2 $

Step 1: Subtract $ x $:
$ 2 = 4x - 2 $

Step 2: Add 2:
$ 4 = 4x $

Step 3: Divide by 4:
$ x = 1 $

✔ Answer: $ x = 1 $

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(i) $ 6x - 9 = 4x - 1 $

Step 1: Subtract $ 4x $:
$ 2x - 9 = -1 $

Step 2: Add 9:
$ 2x = 8 $

Step 3: Divide by 2:
$ x = 4 $

✔ Answer: $ x = 4 $

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(j) $ 5x + 2 = 16 - 2x $

Step 1: Add $ 2x $ to both sides:
$ 7x + 2 = 16 $

Step 2: Subtract 2:
$ 7x = 14 $

Step 3: Divide by 7:
$ x = 2 $

✔ Answer: $ x = 2 $

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(k) $ 3x - 1 = 23 - x $

Step 1: Add $ x $ to both sides:
$ 4x - 1 = 23 $

Step 2: Add 1:
$ 4x = 24 $

Step 3: Divide by 4:
$ x = 6 $

✔ Answer: $ x = 6 $

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(l) $ 6x + 8 = 38 - 4x $

Step 1: Add $ 4x $:
$ 10x + 8 = 38 $

Step 2: Subtract 8:
$ 10x = 30 $

Step 3: Divide by 10:
$ x = 3 $

✔ Answer: $ x = 3 $

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(m) $ 80 - x = 8x - 1 $

Step 1: Add $ x $ to both sides:
$ 80 = 9x - 1 $

Step 2: Add 1:
$ 81 = 9x $

Step 3: Divide by 9:
$ x = 9 $

✔ Answer: $ x = 9 $

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(n) $ 2x + 7 = 17 - 8x $

Step 1: Add $ 8x $:
$ 10x + 7 = 17 $

Step 2: Subtract 7:
$ 10x = 10 $

Step 3: Divide by 10:
$ x = 1 $

✔ Answer: $ x = 1 $

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(o) $ 15 - x = 27 - 3x $

Step 1: Add $ 3x $:
$ 15 + 2x = 27 $

Step 2: Subtract 15:
$ 2x = 12 $

Step 3: Divide by 2:
$ x = 6 $

✔ Answer: $ x = 6 $

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(p) $ 12x - 20 = 15x - 38 $

Step 1: Subtract $ 12x $:
$ -20 = 3x - 38 $

Step 2: Add 38:
$ 18 = 3x $

Step 3: Divide by 3:
$ x = 6 $

✔ Answer: $ x = 6 $

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(q) $ 35x + 10 = 20x + 175 $

Step 1: Subtract $ 20x $:
$ 15x + 10 = 175 $

Step 2: Subtract 10:
$ 15x = 165 $

Step 3: Divide by 15:
$ x = 11 $

✔ Answer: $ x = 11 $

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(r) $ 14x = 2x + 60 $

Step 1: Subtract $ 2x $:
$ 12x = 60 $

Step 2: Divide by 12:
$ x = 5 $

✔ Answer: $ x = 5 $

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✔ Final Answers:

| Equation | Solution |
|--------|---------|
| (a) | $ x = 3 $ |
| (b) | $ x = 6 $ |
| (c) | $ x = 4 $ |
| (d) | $ x = 9 $ |
| (e) | $ x = 4 $ |
| (f) | $ x = 17 $ |
| (g) | $ x = 8 $ |
| (h) | $ x = 1 $ |
| (i) | $ x = 4 $ |
| (j) | $ x = 2 $ |
| (k) | $ x = 6 $ |
| (l) | $ x = 3 $ |
| (m) | $ x = 9 $ |
| (n) | $ x = 1 $ |
| (o) | $ x = 6 $ |
| (p) | $ x = 6 $ |
| (q) | $ x = 11 $ |
| (r) | $ x = 5 $ |

Let me know if you'd like a summary or explanation for any specific one!