Are the lines given by the equations $$ -4 x+y=5 $$ and $$ -x+4 y=12 $$ parallel, perpendicular, or neither? Why?

Question

UpStudy AI · Accepted Answer

To determine whether the lines given by the equations $$ -4x + y = 5 $$ and $$ -x + 4y = 12 $$ are parallel, perpendicular, or neither, we can analyze their slopes.

1. **Find the slope of the first line ($$ -4x + y = 5 $$):**
   $$
   y = 4x + 5
   $$
   - **Slope ($$ m_1 $$)**: 4

2. **Find the slope of the second line ($$ -x + 4y = 12 $$):**
   $$
   4y = x + 12 \quad \Rightarrow \quad y = \frac{1}{4}x + 3
   $$
   - **Slope ($$ m_2 $$)**: $$ \frac{1}{4} $$

3. **Compare the slopes:**
   - **Parallel Lines:** Slopes are equal.
     - Here, $$ m_1 = 4 $$ and $$ m_2 = \frac{1}{4} $$. Since $$ 4 \neq \frac{1}{4} $$, the lines are **not parallel**.
   
   - **Perpendicular Lines:** Slopes are negative reciprocals (i.e., $$ m_1 \times m_2 = -1 $$).
     - $$ 4 \times \frac{1}{4} = 1 $$, which is **not equal** to $$-1$$. Therefore, the lines are **not perpendicular**.

**Conclusion:**
The two lines are **neither parallel nor perpendicular**.

**Answer:**  
They are neither parallel nor perpendicular because their slopes are different and not negative reciprocals.
The lines are neither parallel nor perpendicular because their slopes are different and not negative reciprocals.

Are the lines given by the equations \( -4 x+y=5 \) and \( -x+4 y=12 \) parallel, perpendicular, or neither? Why?

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Geometry Questions