The equations of three lines are given below.
Line 1:
Line 2:
Line
For each pair of lines, determine whether they are parallel, perpendicular, or neither:
Line 1 and Line 2: Parallel Perpendicular o Neither
Line 1 and Line 3: Parallel OPerpendicular Neither 2 and Line 3: Parallel Perpendicular O Neither

Ask by Martin Estrada. in the United States

Mar 31,2025

Question

The equations of three lines are given below.
Line 1:
Line 2:
Line
For each pair of lines, determine whether they are parallel, perpendicular, or neither:
Line 1 and Line 2: Parallel Perpendicular o Neither
Line 1 and Line 3: Parallel OPerpendicular Neither 2 and Line 3: Parallel Perpendicular O Neither

Ask by Martin Estrada. in the United States

Mar 31,2025

UpStudy AI · Accepted Answer

1. **Determine the slope of each line:**

- **Line 1:** $$ y = -4x - 5 $$

The slope is $$ m_1 = -4 $$.

- **Line 2:** $$ y = \frac{1}{4}x + 2 $$

The slope is $$ m_2 = \frac{1}{4} $$.

- **Line 3:** $$ 4x + 16y = -16 $$

Solve for $$ y $$:

$$
     16y = -4x - 16
     $$
     
     $$
     y = -\frac{4}{16}x - 1 = -\frac{1}{4}x - 1
     $$

The slope is $$ m_3 = -\frac{1}{4} $$.

2. **Compare pairs of lines:**

- **Line 1 and Line 2:**

Check if the slopes are negative reciprocals:
     
     $$
     m_1 \cdot m_2 = (-4) \cdot \frac{1}{4} = -1
     $$
     
     Since $$ m_1 \cdot m_2 = -1 $$, these lines are **perpendicular**.

- **Line 1 and Line 3:**

Compare the slopes:
     
     $$
     m_1 = -4, \quad m_3 = -\frac{1}{4}
     $$
     
     They are not equal, so the lines are not parallel. Check for perpendicularity:
     
     $$
     m_1 \cdot m_3 = (-4) \cdot \left(-\frac{1}{4}\right) = 1
     $$
     
     Since $$ m_1 \cdot m_3 \neq -1 $$, these lines are **neither** parallel nor perpendicular.

- **Line 2 and Line 3:**

Compare the slopes:

$$
     m_2 = \frac{1}{4}, \quad m_3 = -\frac{1}{4}
     $$
     
     They are not equal, so the lines are not parallel. Check for perpendicularity:
     
     $$
     m_2 \cdot m_3 = \frac{1}{4} \cdot \left(-\frac{1}{4}\right) = -\frac{1}{16}
     $$
     
     Since $$ m_2 \cdot m_3 \neq -1 $$, these lines are **neither** parallel nor perpendicular.

3. **Final Answers:**

- **Line 1 and Line 2:** Perpendicular
   - **Line 1 and Line 3:** Neither
   - **Line 2 and Line 3:** Neither
- **Line 1 and Line 2:** Perpendicular - **Line 1 and Line 3:** Neither - **Line 2 and Line 3:** Neither

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