Question

Determine whether each pair of lines is parallel, perpendicular or neither.

A. y = 3x -9; y = -1/3x +2

B. y = 9/7x - 19/7; y -1 = 9/7(x+3)

C. x = -3; x = 4

Answer 1

To determine whether each pair of lines is parallel, perpendicular, or neither, we should compare their slopes (m).

**A.**
Lines: \( y = 3x - 9 \) and \( y = -\frac{1}{3}x + 2 \)
- The slope of the first line \( y = 3x - 9 \) is \( m_1 = 3 \).
- The slope of the second line \( y = -\frac{1}{3}x + 2 \) is \( m_2 = -\frac{1}{3} \).

To check if the lines are parallel or perpendicular:
- Lines are parallel if their slopes are equal: \( m_1 \neq m_2 \) (Not parallel).
- Lines are perpendicular if the product of their slopes is -1:
\[ m_1 \times m_2 = 3 \times -\frac{1}{3} = -1 \] (Perpendicular).

**Conclusion: Lines A are perpendicular.**

---

**B.**
Lines: \( y = \frac{9}{7}x - \frac{19}{7} \) and \( y - 1 = \frac{9}{7}(x + 3) \)
- The slope of the first line \( y = \frac{9}{7}x - \frac{19}{7} \) is \( m_1 = \frac{9}{7} \).
- The second equation can be rewritten as follows:
\[ y - 1 = \frac{9}{7}x + \frac{27}{7} \]
\[ y = \frac{9}{7}x + \frac{27}{7} + 1 = \frac{9}{7}x + \frac{27}{7} + \frac{7}{7} = \frac{9}{7}x + \frac{34}{7} \]
So, the slope of the second line is \( m_2 = \frac{9}{7} \).

To check if the lines are parallel or perpendicular:
- Lines are parallel if their slopes are equal: \( m_1 = m_2 = \frac{9}{7} \) (Parallel).
- Lines are perpendicular if the product of their slopes is -1:
\[ m_1 \times m_2 = \frac{9}{7} \times \frac{9}{7} \neq -1 \] (Not perpendicular).

**Conclusion: Lines B are parallel.**

---

**C.**
Lines: \( x = -3 \) and \( x = 4 \)
- Both lines are vertical lines. The slope of a vertical line is undefined.
- Since both lines are vertical, they are parallel to each other.

**Conclusion: Lines C are parallel.**

---

### Final summary:
- **A:** Perpendicular
- **B:** Parallel
- **C:** Parallel