Given two lines with slopes of $$ m_1 = 5 $$ and $$ m_2 = -\frac{1}{5} $$, determine if the lines are parallel, perpendicular, or neither.

Question

UpStudy AI · Accepted Answer

To determine whether the two lines are **parallel**, **perpendicular**, or **neither**, we can analyze their slopes.

Given:
- **Slope of the first line ($$ m_1 $$)**: 5
- **Slope of the second line ($$ m_2 $$)**: $$-\frac{1}{5}$$

### 1. **Checking for Parallel Lines**
Two lines are **parallel** if and only if their slopes are **equal**.

$$
m_1 = m_2 \quad \text{?}
$$

$$
5 \neq -\frac{1}{5}
$$

Since $$ 5 $$ is not equal to $$-\frac{1}{5}$$, the lines are **not parallel**.

### 2. **Checking for Perpendicular Lines**
Two lines are **perpendicular** if and only if the product of their slopes is **-1**.

$$
m_1 \times m_2 = -1 \quad \text{?}
$$

$$
5 \times \left(-\frac{1}{5}\right) = -1
$$

$$
-1 = -1
$$

Since the product of the slopes is $$-1$$, the lines are **perpendicular**.

### **Conclusion**
The two lines with slopes $$ m_1 = 5 $$ and $$ m_2 = -\frac{1}{5} $$ are **perpendicular**.
The lines are perpendicular.

Given two lines with slopes of \( m_1 = 5 \) and \( m_2 = -\frac{1}{5} \), determine if the lines are parallel, perpendicular, or neither.