Question

Determine if the following pairs of lines are intersecting, perpendicular or parallel
Pair A; 2x - y = 7 and 3x - 4y = -8
Pair B; y - 6 = 1/4 (x+2) and y = -4x - 10
Pair C; y = -2/3x + 7 and -2x - 3y = 12

Answer 1

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

### Pair A:
Equations:
1. \( 2x - y = 7 \)
2. \( 3x - 4y = -8 \)

**Convert to slope-intercept form (y = mx + b)**

1. For \( 2x - y = 7 \):
\[
-y = -2x + 7 \implies y = 2x - 7
\]
(Slope, \( m_1 = 2 \))

2. For \( 3x - 4y = -8 \):
\[
-4y = -3x - 8 \implies y = \frac{3}{4}x + 2
\]
(Slope, \( m_2 = \frac{3}{4} \))

**Compare slopes:**
- \( m_1 = 2 \) and \( m_2 = \frac{3}{4} \)

Since the slopes are different, the lines are **intersecting**.

### Pair B:
Equations:
1. \( y - 6 = \frac{1}{4}(x + 2) \)
2. \( y = -4x - 10 \)

**Convert to slope-intercept form:**

1. Rearranging \( y - 6 = \frac{1}{4}(x + 2) \):
\[
y - 6 = \frac{1}{4}x + \frac{1}{2} \implies y = \frac{1}{4}x + 6.5
\]
(Slope, \( m_1 = \frac{1}{4} \))

2. The second equation is already in slope-intercept form \( y = -4x - 10 \):
(Slope, \( m_2 = -4 \))

**Compare slopes:**
- \( m_1 = \frac{1}{4} \) and \( m_2 = -4 \)

Since the slopes are different and \( m_1 \cdot m_2 = \frac{1}{4} \cdot (-4) = -1 \), the lines are **perpendicular**.

### Pair C:
Equations:
1. \( y = -\frac{2}{3}x + 7 \)
2. \( -2x - 3y = 12 \)

**Convert to slope-intercept form:**

1. The first line is already in slope-intercept form:
(Slope, \( m_1 = -\frac{2}{3} \))

2. Rearranging \( -2x - 3y = 12 \):
\[
-3y = 2x + 12 \implies y = -\frac{2}{3}x - 4
\]
(Slope, \( m_2 = -\frac{2}{3} \))

**Compare slopes:**
- \( m_1 = -\frac{2}{3} \) and \( m_2 = -\frac{2}{3} \)

Since the slopes are equal, the lines are **parallel**.

### Summary:
- **Pair A:** Intersecting
- **Pair B:** Perpendicular
- **Pair C:** Parallel