Question

Are the pairs of lines parallel, perpendicular, or intersecting?(3 points)
y−6=14(x+2)
y=−4x−10

2x−y=7
3x−4y=−8

y=−23x+7
−2x−3y=12
answers
parellel, perpendiculer, intersecting

Answer 1

To determine whether the pairs of lines are parallel, perpendicular, or intersecting, we can convert the equations into the slope-intercept form \(y = mx + b\) where \(m\) is the slope.

### Pair 1
1. **First Line:**
\(y - 6 = 14(x + 2)\)
Expand:
\(y - 6 = 14x + 28\)
Therefore,
\(y = 14x + 34\)
**Slope:** \(m_1 = 14\)

2. **Second Line:**
\(y = -4x - 10\)
**Slope:** \(m_2 = -4\)

### Analysis of Pair 1
- The slopes are \(m_1 = 14\) and \(m_2 = -4\).
- Since the product of the slopes \(m_1 \cdot m_2 = 14 \cdot (-4) = -56\), which is negative, the lines are **perpendicular**.

### Pair 2
1. **First Line:**
\(2x - y = 7\)
Rearranging gives:
\(y = 2x - 7\)
**Slope:** \(m_1 = 2\)

2. **Second Line:**
\(3x - 4y = -8\)
Rearranging gives:
\(4y = 3x + 8 \implies y = \frac{3}{4}x + 2\)
**Slope:** \(m_2 = \frac{3}{4}\)

### Analysis of Pair 2
- The slopes are \(m_1 = 2\) and \(m_2 = \frac{3}{4}\).
- Since the product of the slopes \(m_1 \cdot m_2 = 2 \cdot \frac{3}{4} = \frac{3}{2}\), which is positive, the lines are **intersecting**.

### Pair 3
1. **First Line:**
\(y = -23x + 7\)
**Slope:** \(m_1 = -23\)

2. **Second Line:**
\(-2x - 3y = 12\)
Rearranging gives:
\(-3y = 2x + 12 \implies y = -\frac{2}{3}x - 4\)
**Slope:** \(m_2 = -\frac{2}{3}\)

### Analysis of Pair 3
- The slopes are \(m_1 = -23\) and \(m_2 = -\frac{2}{3}\).
- Since the slopes have the same sign and are not equal (not parallel), the lines are **intersecting**.

### Summary of the results:
- Pair 1: **Perpendicular**
- Pair 2: **Intersecting**
- Pair 3: **Intersecting**