Question

Are the pairs of lines parallel, perpendicular, or intersecting
2x - y = 7
3x - 4y = -8

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

y= - 2/3 x + 7
-2x - 3y = 12
Intersecting. Perpendicular. Parallel

Answer 1

To determine whether the pairs of lines are parallel, perpendicular, or intersecting, we first need to find their slopes. If the slopes are equal, the lines are parallel; if the product of the slopes is -1, they are perpendicular; if neither condition is met, they intersect.

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

**Finding slopes**:

1. Rearrange the first equation into \( y = mx + b \) form:
\[
y = 2x - 7
\]
Slope \( m_1 = 2 \)

2. Rearrange the second equation:
\[
3x - 4y = -8 \implies -4y = -3x - 8 \implies y = \frac{3}{4}x + 2
\]
Slope \( m_2 = \frac{3}{4} \)

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

Since \( m_1 \) and \( m_2 \) are not equal and their product (\(2 \times \frac{3}{4} = \frac{3}{2}\)) is not -1, the lines are **intersecting**.

### Pair 2:
**Equations**:
1. \( y - 6 = \frac{1}{4}(x + 2) \)
2. \( y = -6 = \frac{1}{4}(x + 2) \) (This appears to be incorrectly stated; I’ll assume the equation was meant to be \( y + 6 = \frac{1}{4}(x + 2) \))

**Finding slopes**:

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

2. Rearranging the second (correcting the potential mistake):
\[
y + 6 = \frac{1}{4}(x + 2) \implies y = \frac{1}{4}x - 6 + \frac{1}{2} \implies y = \frac{1}{4}x - 5.5
\]
Slope \( m_2 = \frac{1}{4} \)

**Checking the slopes**:
- \( m_1 = \frac{1}{4} \)
- \( m_2 = \frac{1}{4} \)

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

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

**Finding slopes**:

1. Slope is given directly in the first equation: \( m_1 = -\frac{2}{3} \)

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

**Checking the slopes**:
- \( m_1 = -\frac{2}{3} \)
- \( m_2 = -\frac{2}{3} \)

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

### Summary
- Pair 1: **Intersecting**
- Pair 2: **Parallel**
- Pair 3: **Parallel**