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Let's go over the equations shown in your homework. I’ll provide detailed steps for solving each one. These problems involve solving linear equations with variables on both sides.

### 1) \(-x + 7 = 5x + 7 - 6x\)
- Combine like terms on the right side: \(5x - 6x = -x\)
- The equation becomes \(-x + 7 = -x + 7\)
- Subtract \(-x\) from both sides: \(7 = 7\)
- This equation is an **identity**, meaning it's true for all values of \(x\).

### 2) \(8 + x - 6 + 3 = x + 5\)
- Simplify the left side: \(8 - 6 + 3 = 5\), so the equation becomes \(5 + x = x + 5\)
- Subtract \(x\) from both sides: \(5 = 5\)
- This is also an **identity**, so it's true for all \(x\).

### 3) \(-9 - x = -x - 5\)
- Add \(x\) to both sides: \(-9 = -5\)
- This is a **contradiction** (since \(-9\) is not equal to \(-5\)), meaning there is **no solution**.

### 4) \(1 + 3v = 11 - 6 + 4v - 7\)
- Simplify both sides: 
  \[
  1 + 3v = -2 + 4v
  \]
- Subtract \(3v\) from both sides: \(1 = -2 + v\)
- Add 2 to both sides: \(3 = v\)
- So, \(v = 3\).

### 5) \(6 - 5x = 1 - 4x\)
- Add \(5x\) to both sides: \(6 = 1 + x\)
- Subtract 1 from both sides: \(5 = x\)
- So, \(x = 5\).

### 6) \(3v - 2 = -4 + 3v\)
- Subtract \(3v\) from both sides: \(-2 = -4\)
- This is a **contradiction**, so there is **no solution**.

### 7) \(16 - 3 + 7x - 3x = 6x - 1\)
- Simplify both sides: \(13 + 4x = 6x - 1\)
- Subtract \(4x\) from both sides: \(13 = 2x - 1\)
- Add 1 to both sides: \(14 = 2x\)
- Divide by 2: \(x = 7\).

### 8) \(-6n + 7n = -12 - 4n + 2n\)
- Combine like terms: \(n = -12 - 2n\)
- Add \(2n\) to both sides: \(3n = -12\)
- Divide by 3: \(n = -4\).

### 9) \(r + 3 + 4 + 11 = -r - 4r\)
- Simplify both sides: \(r + 18 = -5r\)
- Add \(5r\) to both sides: \(6r + 18 = 0\)
- Subtract 18 from both sides: \(6r = -18\)
- Divide by 6: \(r = -3\).

### 10) \(-9 + 8n = 4n + 3\)
- Subtract \(4n\) from both sides: \(-9 + 4n = 3\)
- Add 9 to both sides: \(4n = 12\)
- Divide by 4: \(n = 3\).

Would you like further details or explanations for any of these solutions?

Here are 5 related questions to practice:

1. Solve: \(2x + 3 = 7x - 9\)
2. Solve: \(4 - 2n = 3n + 5\)
3. Solve: \(5x - 7 = 3x + 9 - 4x\)
4. Solve: \(6y + 2 - 3y = y - 4\)
5. Solve: \(8z + 3 = 4z + 19\)

**Tip:** Always combine like terms on each side of the equation before moving variables or constants across the equal sign.

Learn how to solve linear equations with variables on both sides from a Grade 8 homework sheet. This example covers key algebra concepts such as solving identities, contradictions, and simplifying terms. Perfect for students in Grades 8-9.

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