Question

Y=4/7×-5
Y=-7/4×-1
parallel


perpendicular


intersecting but not perpendicular

Answer 1

To determine the relationship between the two lines represented by the equations \( Y = \frac{4}{7}X - 5 \) and \( Y = -\frac{7}{4}X - 1 \), we need to analyze their slopes:

1. **Slope of the first line** (\( Y = \frac{4}{7}X - 5 \)):
- The slope \( m_1 \) is \( \frac{4}{7} \).

2. **Slope of the second line** (\( Y = -\frac{7}{4}X - 1 \)):
- The slope \( m_2 \) is \( -\frac{7}{4} \).

### Relationship of slopes:

- **Parallel lines**: Two lines are parallel if their slopes are equal (i.e., \( m_1 = m_2 \)).
- **Perpendicular lines**: Two lines are perpendicular if the product of their slopes is -1 (i.e., \( m_1 \cdot m_2 = -1 \)).
- **Intersecting but not perpendicular**: Two lines intersect at a point and are not parallel or perpendicular.

### Checking the slopes:

- To check if they are perpendicular:
\[
m_1 \cdot m_2 = \frac{4}{7} \cdot \left(-\frac{7}{4}\right) = -1.
\]
Since \( m_1 \cdot m_2 = -1 \), the lines are perpendicular.

### Summary:
- The relationship between the two lines is that they are **perpendicular**.