Question

Are the pairs of lines parallel, perpendicular, or intersecting?(3 points) Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse. 2x−y=7 3x−4y=−8 y−6=14(x+2) y=−4x−10 y=−23x+7 −2x−3y=12

Answer 1

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

1. **Convert equations to slope-intercept form:**

- For **2x - y = 7**:
\[
-y = -2x + 7 \implies y = 2x - 7 \quad (m_1 = 2)
\]

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

- For **y - 6 = 14(x + 2)**:
\[
y - 6 = 14x + 28 \implies y = 14x + 34 \quad (m_3 = 14)
\]

- For **y = -4x - 10**:
\[
(m_4 = -4)
\]

- For **y = -23x + 7**:
\[
(m_5 = -23)
\]

- For **-2x - 3y = 12**:
\[
-3y = 2x + 12 \implies y = -\frac{2}{3}x - 4 \quad (m_6 = -\frac{2}{3})
\]

2. **Analyze the slopes for parallelism and perpendicularity**:

- Lines are **parallel** if their slopes (m) are equal.
- Lines are **perpendicular** if the product of their slopes is -1 (i.e., \(m_1 \cdot m_2 = -1\)).

3. **Compare the slopes**:

- **Lines 1 and 2**: \( m_1 = 2 \) and \( m_2 = \frac{3}{4} \) (Intersecting)
- **Lines 1 and 3**: \( m_1 = 2 \) and \( m_3 = 14 \) (Intersecting)
- **Lines 1 and 4**: \( m_1 = 2 \) and \( m_4 = -4 \) (Intersecting)
- **Lines 1 and 5**: \( m_1 = 2 \) and \( m_5 = -23 \) (Intersecting)
- **Lines 1 and 6**: \( m_1 = 2 \) and \( m_6 = -\frac{2}{3} \) (Intersecting)

- **Lines 2 and 3**: \( m_2 = \frac{3}{4} \) and \( m_3 = 14 \) (Intersecting)
- **Lines 2 and 4**: \( m_2 = \frac{3}{4} \) and \( m_4 = -4 \) (Intersecting)
- **Lines 2 and 5**: \( m_2 = \frac{3}{4} \) and \( m_5 = -23 \) (Intersecting)
- **Lines 2 and 6**: \( m_2 = \frac{3}{4} \) and \( m_6 = -\frac{2}{3} \) (Intersecting)

- **Lines 3 and 4**: \( m_3 = 14 \) and \( m_4 = -4 \) (Intersecting)
- **Lines 3 and 5**: \( m_3 = 14 \) and \( m_5 = -23 \) (Intersecting)
- **Lines 3 and 6**: \( m_3 = 14 \) and \( m_6 = -\frac{2}{3} \) (Intersecting)

- **Lines 4 and 5**: \( m_4 = -4 \) and \( m_5 = -23 \) (Intersecting)
- **Lines 4 and 6**: \( m_4 = -4 \) and \( m_6 = -\frac{2}{3} \) (Intersecting)

- **Lines 5 and 6**: \( m_5 = -23 \) and \( m_6 = -\frac{2}{3} \) (Intersecting)

**Summary of pairs**: All pairs of the given lines are intersecting since no two lines have the same slope and there are no pairs that are perpendicular (the product of their slopes does not equal -1).

**Overall conclusion**: All pairs are **intersecting**.