Question

Are the pairs of lines parallel, perpendicular, or intersecting?:
y - 6 = 1/4 (x - 2), y = -4x - 10
2x - y = 7, 3x - 4y = -8
y = - 2/3 x + 7, -2x - 3y = 12

Answer 1

To determine whether the pairs of lines are parallel, perpendicular, or intersecting, we first need to convert each equation into the slope-intercept form \( y = mx + b \), where \( m \) represents the slope.

### Pair 1:
1. **Equation 1**: \( y - 6 = \frac{1}{4}(x - 2) \)

We can rewrite this in slope-intercept form:
\[
y - 6 = \frac{1}{4}x - \frac{1}{2}
\]
\[
y = \frac{1}{4}x + 5.5
\]
**Slope \( m_1 = \frac{1}{4} \)**

2. **Equation 2**: \( y = -4x - 10 \)

**Slope \( m_2 = -4 \)**

To check if they are parallel or perpendicular:
- Two lines are **parallel** if their slopes are equal: \( m_1 = m_2 \).
- Two lines are **perpendicular** if the product of their slopes is \(-1\): \( m_1 \cdot m_2 = -1 \).

Calculating the product:
\[
\frac{1}{4} \cdot (-4) = -1
\]
Since their slopes satisfy the condition for being perpendicular, these two lines are **perpendicular**.

### Pair 2:
1. **Equation 1**: \( 2x - y = 7 \)

To get the slope-intercept form:
\[
-y = -2x + 7
\]
\[
y = 2x - 7
\]
**Slope \( m_1 = 2 \)**

2. **Equation 2**: \( 3x - 4y = -8 \)

Rearranging:
\[
-4y = -3x - 8
\]
\[
y = \frac{3}{4}x + 2
\]
**Slope \( m_2 = \frac{3}{4} \)**

Checking for parallel or perpendicular:
- Product of slopes:
\[
2 \cdot \frac{3}{4} = \frac{3}{2}
\]
Since the slopes are not equal and their product is not \(-1\), these lines are **intersecting**.

### Pair 3:
1. **Equation 1**: \( y = -\frac{2}{3}x + 7 \)

**Slope \( m_1 = -\frac{2}{3} \)**

2. **Equation 2**: \( -2x - 3y = 12 \)

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

Checking for parallel or perpendicular:
- Product of slopes:
\[
-\frac{2}{3} \cdot -\frac{2}{3} = \frac{4}{9}
\]
Since the slopes are equal, these lines are **parallel**.

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