Worksheet for solving linear equations featuring eight problems with variables, parentheses, and fractions.

Let's solve each of the linear equations step by step and explain the process.

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1. $ 2(m + 6) = 48 $

Step 1: Distribute the 2 on the left side:
$$
2m + 12 = 48
$$

Step 2: Subtract 12 from both sides:
$$
2m = 36
$$

Step 3: Divide both sides by 2:
$$
m = 18
$$

✔ Answer: $ m = 18 $

---

2. $ 8x + 9 - 3x = 8 + 4x + 5 $

Step 1: Combine like terms on both sides:

Left: $ 8x - 3x + 9 = 5x + 9 $
Right: $ 8 + 5 + 4x = 13 + 4x $

So:
$$
5x + 9 = 4x + 13
$$

Step 2: Subtract $ 4x $ from both sides:
$$
x + 9 = 13
$$

Step 3: Subtract 9 from both sides:
$$
x = 4
$$

✔ Answer: $ x = 4 $

---

3. $ 8a - (4a + 32) = 16 $

Step 1: Distribute the negative sign:
$$
8a - 4a - 32 = 16
$$

Step 2: Combine like terms:
$$
4a - 32 = 16
$$

Step 3: Add 32 to both sides:
$$
4a = 48
$$

Step 4: Divide by 4:
$$
a = 12
$$

✔ Answer: $ a = 12 $

---

4. $ -3(x - 1) + 8(x - 3) = 6x + 7 - 5x $

Step 1: Expand both sides:

Left:
$$
-3x + 3 + 8x - 24 = ( -3x + 8x ) + (3 - 24) = 5x - 21
$$

Right:
$$
6x - 5x + 7 = x + 7
$$

Now equation is:
$$
5x - 21 = x + 7
$$

Step 2: Subtract $ x $ from both sides:
$$
4x - 21 = 7
$$

Step 3: Add 21 to both sides:
$$
4x = 28
$$

Step 4: Divide by 4:
$$
x = 7
$$

✔ Answer: $ x = 7 $

---

5. $ 10x + 5(5x - 3) = 5(6x + 4) $

Step 1: Expand both sides:

Left:
$$
10x + 25x - 15 = 35x - 15
$$

Right:
$$
30x + 20
$$

Now:
$$
35x - 15 = 30x + 20
$$

Step 2: Subtract $ 30x $ from both sides:
$$
5x - 15 = 20
$$

Step 3: Add 15 to both sides:
$$
5x = 35
$$

Step 4: Divide by 5:
$$
x = 7
$$

✔ Answer: $ x = 7 $

---

6. $ 0.02m + 0.08(8 - m) = 1.78 $

Step 1: Expand the parentheses:
$$
0.02m + 0.08 \cdot 8 - 0.08m = 1.78
$$
$$
0.02m + 0.64 - 0.08m = 1.78
$$

Step 2: Combine like terms:
$$
-0.06m + 0.64 = 1.78
$$

Step 3: Subtract 0.64 from both sides:
$$
-0.06m = 1.14
$$

Step 4: Divide by -0.06:
$$
m = \frac{1.14}{-0.06} = -19
$$

✔ Answer: $ m = -19 $

---

7. $ \frac{1}{3}(10 - 2x) = \frac{1}{3}(8 - 2x) $

Step 1: Multiply both sides by 3 to eliminate denominators:
$$
10 - 2x = 8 - 2x
$$

Step 2: Add $ 2x $ to both sides:
$$
10 = 8
$$

✘ This is not true — contradiction.

This means no solution exists.

✔ Answer: No solution (inconsistent equation)

---

8. $ -14 + 6b + 7 - 2b = 1 + 5b $

Step 1: Combine like terms on the left:

Constants: $ -14 + 7 = -7 $
Variables: $ 6b - 2b = 4b $

So:
$$
4b - 7 = 1 + 5b
$$

Step 2: Subtract $ 5b $ from both sides:
$$
- b - 7 = 1
$$

Step 3: Add 7 to both sides:
$$
- b = 8
$$

Step 4: Multiply both sides by -1:
$$
b = -8
$$

✔ Answer: $ b = -8 $

---

✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | $ m = 18 $ |
| 2 | $ x = 4 $ |
| 3 | $ a = 12 $ |
| 4 | $ x = 7 $ |
| 5 | $ x = 7 $ |
| 6 | $ m = -19 $ |
| 7 | No solution |
| 8 | $ b = -8 $ |

Let me know if you'd like this in a printable format or need explanations for any specific steps!