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Let's solve the given equations using a two-column proof format.

### 1. \( 8x - 10 = 3(6 - 2x) \)

| **Steps** | **Reasons** |
|-----------|-------------|
| \( 8x - 10 = 3(6 - 2x) \) | Given |
| \( 8x - 10 = 18 - 6x \) | Distribute the 3 across \( (6 - 2x) \) |
| \( 8x + 6x = 18 + 10 \) | Add \( 6x \) and \( 10 \) to both sides |
| \( 14x = 28 \) | Simplify |
| \( x = 2 \) | Divide both sides by 14 |

---

### 2. \( 5n - 42 = 12n \)

| **Steps** | **Reasons** |
|-----------|-------------|
| \( 5n - 42 = 12n \) | Given |
| \( 5n - 12n = 42 \) | Subtract \( 12n \) from both sides |
| \( -7n = 42 \) | Simplify |
| \( n = -6 \) | Divide both sides by -7 |

---

### 3. \( \frac{m}{-3} + 10 = -1 \)

| **Steps** | **Reasons** |
|-----------|-------------|
| \( \frac{m}{-3} + 10 = -1 \) | Given |
| \( \frac{m}{-3} = -1 - 10 \) | Subtract 10 from both sides |
| \( \frac{m}{-3} = -11 \) | Simplify |
| \( m = 33 \) | Multiply both sides by -3 |

---

### 4. \( 3x - 2 = 2(x + 5) \)

| **Steps** | **Reasons** |
|-----------|-------------|
| \( 3x - 2 = 2(x + 5) \) | Given |
| \( 3x - 2 = 2x + 10 \) | Distribute the 2 over \( (x + 5) \) |
| \( 3x - 2x = 10 + 2 \) | Subtract \( 2x \) and add 2 to both sides |
| \( x = 12 \) | Simplify |

---

Would you like more details on any step? Here are five related questions:

1. How do you verify a solution for an equation?
2. Can you explain how distributing works in algebra?
3. What is the reason for combining like terms in an equation?
4. How can you solve equations involving fractions?
5. How do you handle equations where variables appear on both sides?

**Tip:** Always perform the same operation on both sides of an equation to maintain balance!

Write a two-column proof for the following equations:
1) 8x - 10 = 3(6 - 2x)
2) 5n - 42 = 12n
3) m/-3 + 10 = -1
4) 3x - 2 = 2(x + 5)

Learn how to solve linear equations using a two-column proof method. The equations involve applying the distributive property and combining like terms to isolate variables. Suitable for students in Grades 7-9.

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