Question

Are the pairs of lines parallel, perpendicular, or intersecting? (3 points)

2x - y = 7

3x - 4y = - 8

y - 6 = 1/4 * (x + 2)

y = - 4x - 10

y = - 2/3 * x + 7

- 2x - 3y = 12

# Intersecting

>

#Perpendicular

# Parallel

Answer 1

To determine the relationship between the pairs of lines given by their equations, we can convert all of the equations to slope-intercept form (y = mx + b), where m represents the slope. Two lines are **parallel** if they have the same slope, **perpendicular** if the product of their slopes is -1, and **intersecting** if neither condition applies.

Let's find the slopes for each pair of lines.

1. **Different pairs of lines**:

- **First Pair**:
1. \( 2x - y = 7 \)
\[
y = 2x - 7 \quad \text{(slope } m_1 = 2\text{)}
\]

2. \( 3x - 4y = -8 \)
\[
4y = 3x + 8 \implies y = \frac{3}{4}x + 2 \quad \text{(slope } m_2 = \frac{3}{4}\text{)}
\]

- Since \( m_1 \neq m_2 \) and \( m_1 \cdot m_2 \neq -1 \), these lines are **intersecting**.

- **Second Pair**:
1. \( y - 6 = \frac{1}{4}(x + 2) \)
\[
y - 6 = \frac{1}{4}x + \frac{1}{2} \implies y = \frac{1}{4}x + 6.5 \quad \text{(slope } m_3 = \frac{1}{4}\text{)}
\]

2. \( y = -4x - 10 \)
\[
(slope } m_4 = -4\text{)}
\]

- Since \( m_3 \cdot m_4 = \frac{1}{4} \cdot -4 = -1 \), these lines are **perpendicular**.

- **Third Pair**:
1. \( y = -\frac{2}{3}x + 7 \)
\[
(slope } m_5 = -\frac{2}{3}\text{)}
\]

2. \( -2x - 3y = 12 \)
\[
3y = -2x - 12 \implies y = -\frac{2}{3}x - 4 \quad \text{(slope } m_6 = -\frac{2}{3}\text{)}
\]

- Since \( m_5 = m_6 \), these lines are **parallel**.

In summary:
1. The first pair is **intersecting**.
2. The second pair is **perpendicular**.
3. The third pair is **parallel**.