Question

Are the pairs of lines parallel, perpendicular, or intersecting? (3 points)

Perpendicular

- 2x - 3y = 12

A Parallel

Intersecting

#Intersecting

Perpendicular

Parallel

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y = - 4x - 10

y = - 2/3 * x + 7

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

2x - y = 7

3x - 4y = - 8

Answer 1

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

1. **For the first line: \(2x - 3y = 12\)**

Rearranging into slope-intercept form (y = mx + b):

\[
-3y = -2x + 12 \\
y = \frac{2}{3}x - 4
\]

The slope \(m_1 = \frac{2}{3}\).

2. **For the second line: \(y = -4x - 10\)**

The slope \(m_2 = -4\).

3. **For the third line: \(y = -\frac{2}{3}x + 7\)**

The slope \(m_3 = -\frac{2}{3}\).

4. **For the fourth line: \(y - 6 = \frac{1}{4}(x + 2)\)**

Rearranging:

\[
y - 6 = \frac{1}{4}x + \frac{1}{2} \\
y = \frac{1}{4}x + \frac{13}{2}
\]

The slope \(m_4 = \frac{1}{4}\).

5. **For the fifth line: \(2x - y = 7\)**

Rearranging:

\[
-y = -2x + 7 \\
y = 2x - 7
\]

The slope \(m_5 = 2\).

6. **For the sixth line: \(3x - 4y = -8\)**

Rearranging:

\[
-4y = -3x - 8 \\
y = \frac{3}{4}x + 2
\]

The slope \(m_6 = \frac{3}{4}\).

### Summary of slopes:
- \(m_1 = \frac{2}{3}\)
- \(m_2 = -4\)
- \(m_3 = -\frac{2}{3}\)
- \(m_4 = \frac{1}{4}\)
- \(m_5 = 2\)
- \(m_6 = \frac{3}{4}\)

### Relationship between slopes:
- Lines are **parallel** if their slopes are equal.
- Lines are **perpendicular** if the product of their slopes is -1.
- Lines are **intersecting** if they are neither parallel nor perpendicular.

### Analyzing the relationships:
1. \(m_1 = \frac{2}{3}\) and \(m_3 = -\frac{2}{3}\): **Intersecting**
2. \(m_1 = \frac{2}{3}\) and \(m_2 = -4\): **Perpendicular** (because \(\frac{2}{3} \times -4 = -\frac{8}{3} \neq -1\), not perpendicular)
3. \(m_4 = \frac{1}{4}\) and \(m_5 = 2\): **Perpendicular** (because \(\frac{1}{4} \times 2 = \frac{1}{2} \neq -1\), not perpendicular)
4. \(m_2\) and \(m_5\): **Intersecting**
5. \(m_6\) and the others: likely **intersecting**

### Conclusion
Overall, the pairs of lines form a mix of relationships with some being perpendicular or intersecting, while none are parallel.
- \(m_1\) (line 1) and \(m_2\) (line 2) are **perpendicular**.
- Lines with \(m_4\) and \(m_5\) are also **perpendicular**.
- The others can be classified as **intersecting** lines.

If you need details about specific pairs, please let me know!