Question

Are the pairs of lines parallel, perpendicular, or intersecting?(3 points)
1.) y−6=1/4(x+2)
y=−4x−10

2.) y=−2/3x+7
−2x−3y=12

3.) 2x−y=7
3x−4y=−8

Answer 1

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

**1.) For the lines:**
- Line 1: \( y - 6 = \frac{1}{4}(x + 2) \)
- Rearranging to slope-intercept form (y = mx + b):
\[
y - 6 = \frac{1}{4}(x + 2) \\
y - 6 = \frac{1}{4}x + \frac{1}{2} \\
y = \frac{1}{4}x + 6 + \frac{1}{2} \\
y = \frac{1}{4}x + 6.5
\]
- So, the slope \( m_1 = \frac{1}{4} \).

- Line 2: \( y = -4x - 10 \) is already in slope-intercept form.
- Therefore, \( m_2 = -4 \).

- Now to compare the slopes:
- Since \( m_1 \) and \( m_2 \) are not equal nor are they negative reciprocals, the lines are **intersecting**.

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

- Line 2: Rearranging \( -2x - 3y = 12 \):
\[
-3y = 2x + 12 \\
y = -\frac{2}{3}x - 4
\]
- The slope \( m_2 = -\frac{2}{3} \).

- Now to compare the slopes:
- Since \( m_1 = m_2 \), the lines are **parallel**.

---

**3.) For the lines:**
- Line 1: Rearranging \( 2x - y = 7 \):
\[
-y = -2x + 7 \\
y = 2x - 7
\]
- The slope \( m_1 = 2 \).

- Line 2: Rearranging \( 3x - 4y = -8 \):
\[
-4y = -3x - 8 \\
y = \frac{3}{4}x + 2
\]
- The slope \( m_2 = \frac{3}{4} \).

- Now to compare the slopes:
- Since \( m_1 \) and \( m_2 \) are not equal nor negative reciprocals, the lines are **intersecting**.

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### Summary:
1. Lines: intersecting
2. Lines: parallel
3. Lines: intersecting