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To find which pair of lines are perpendicular, we need to determine their slopes. Two lines are perpendicular if the product of their slopes equals -1.

Let's break down each equation:

1. **\( y = -\frac{1}{3}(1 + x) \)**
   - Simplify: \( y = -\frac{1}{3} - \frac{1}{3}x \)
   - The slope (coefficient of \( x \)) is \( m_1 = -\frac{1}{3} \).

2. **\( y = \frac{x}{3} + 2 \)**
   - The slope is \( m_2 = \frac{1}{3} \).

3. **\( y = 4 + 3x \)**
   - The slope is \( m_3 = 3 \).

4. **\( y = 5 - x \)**
   - Rewrite as \( y = -x + 5 \), so the slope is \( m_4 = -1 \).

### Now check for perpendicularity:

- For perpendicular lines, \( m_1 \times m_2 = -1 \), where \( m_1 \) and \( m_2 \) are the slopes of the two lines.
- \( (-\frac{1}{3}) \times \frac{1}{3} = -\frac{1}{9} \) (not perpendicular).
- \( 3 \times (-1) = -3 \) (not perpendicular).
- Checking \( 3 \) and \( -\frac{1}{3} \):
  - \( 3 \times (-\frac{1}{3}) = -1 \) (these lines are perpendicular).

Thus, the pair of perpendicular lines is:

- **\( y = 4 + 3x \)** and **\( y = -\frac{1}{3}(1 + x) \)**.

Would you like further details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you determine the slope from a given linear equation?
2. Why is the product of the slopes of perpendicular lines always -1?
3. Can two vertical lines ever be perpendicular?
4. What does it mean geometrically for lines to be parallel or perpendicular?
5. How would you find the perpendicular bisector of a line segment?

**Tip**: To check if two lines are perpendicular, always multiply their slopes—if the result is -1, the lines are perpendicular.

Choose the pair of perpendicular lines: y = -1/3(1+x), y = x/3 + 2, y = 4 + 3x, y = 5 - x.

Determine which pair of lines are perpendicular by comparing slopes. Understand the concept of slope and the condition for perpendicularity, where the product of the slopes is -1. This problem is ideal for Grades 8-10, focusing on linear equations and their properties.

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