Question

determine whether the two lines below are parallel, perpendicular or neither.
y= 8x+4
3y+24x+15
perpendicular
neither
parallel

Answer 1

To determine the relationship between the two lines, we need to find their slopes.

1. The first line is given in slope-intercept form:
\( y = 8x + 4 \).
The slope (m) of this line is 8.

2. The second line is given in the form of \( 3y + 24x + 15 = 0 \).
We need to rearrange it into slope-intercept form \( y = mx + b \).

Starting with:
\( 3y + 24x + 15 = 0 \)
We can isolate \( y \):
\( 3y = -24x - 15 \)
\( y = -8x - 5 \)

The slope of the second line is -8.

Now we compare the slopes of the two lines:
- The slope of the first line is 8.
- The slope of the second line is -8.

To determine if the lines are parallel, perpendicular, or neither:
- **Parallel lines** have the same slope.
- **Perpendicular lines** have slopes that are negative reciprocals of each other (i.e., \( m_1 \cdot m_2 = -1 \)).

For the given slopes:
\( 8 \) and \( -8 \)
Calculating the product:
\( 8 \cdot (-8) = -64 \) (not equal to -1)

Since the slopes are not equal and they are not negative reciprocals, the two lines are **neither parallel nor perpendicular**.