Question

Are the pairs of lines parallel, perpendicular, or intersecting?(3 points)
y=−2/3x+7
−2x−3y=12

2x−y=7
3x−4y=−8

y−6=1/4(x+2)
y=−4x−10

Answer 1

To determine whether pairs of lines are parallel, perpendicular, or intersecting, we need to analyze their slopes.

1. For the first pair of lines:

- **Line 1:** \( y = -\frac{2}{3}x + 7 \)
- The slope \( m_1 = -\frac{2}{3} \)

- **Line 2:** Convert to slope-intercept form:
\[ -2x - 3y = 12 \]
\[ -3y = 2x + 12 \]
\[ y = -\frac{2}{3}x - 4 \]
- The slope \( m_2 = -\frac{2}{3} \)

**Conclusion:** Since \( m_1 = m_2 \), the lines are **parallel**.

2. For the second pair of lines:

- **Line 1:** Convert to slope-intercept form:
\[ 2x - y = 7 \]
\[ -y = -2x + 7 \]
\[ y = 2x - 7 \]
- The slope \( m_1 = 2 \)

- **Line 2:** Convert to slope-intercept form:
\[ 3x - 4y = -8 \]
\[ -4y = -3x - 8 \]
\[ y = \frac{3}{4}x + 2 \]
- The slope \( m_2 = \frac{3}{4} \)

**Conclusion:** Since \( m_1 \neq m_2 \) and \( m_1 \cdot m_2 \neq -1 \) (i.e., \( 2 \cdot \frac{3}{4} = \frac{3}{2} \)), the lines are **intersecting**.

3. For the third pair of lines:

- **Line 1:** \( y - 6 = \frac{1}{4}(x + 2) \)
- Convert to slope-intercept form:
\[ y - 6 = \frac{1}{4}x + \frac{1}{2} \]
\[ y = \frac{1}{4}x + \frac{1}{2} + 6 \]
\[ y = \frac{1}{4}x + \frac{13}{2} \]
- The slope \( m_1 = \frac{1}{4} \)

- **Line 2:** \( y = -4x - 10 \)
- The slope \( m_2 = -4 \)

**Conclusion:** The slopes \( m_1 \) and \( m_2 \) are not equal (i.e., \( m_1 \cdot m_2 = \frac{1}{4} \cdot -4 = -1 \)), indicating that the lines are **perpendicular**.

### Summary of Results:
1. First pair: **Parallel**
2. Second pair: **Intersecting**
3. Third pair: **Perpendicular**