Question

Are the Lines Parallel, Perpendicular, or Intersecting? y=1/3x+2 y+6x=4 (1 point) Responses Parallel Parallel Perpendicular Perpendicular Intersecting

Answer 1

To determine the relationship between the two lines given by the equations \(y = \frac{1}{3}x + 2\) and \(y + 6x = 4\), we should first identify their slopes.

1. The first equation is in slope-intercept form \(y = mx + b\), where \(m\) is the slope.
- The slope of the first line is \(\frac{1}{3}\).

2. For the second equation \(y + 6x = 4\), we can rearrange it into slope-intercept form:
\[
y = -6x + 4
\]
- The slope of the second line is \(-6\).

Now, we can compare the slopes:
- The slope of the first line is \(\frac{1}{3}\).
- The slope of the second line is \(-6\).

To determine if the lines are parallel, perpendicular, or intersecting:
- **Parallel lines** have the same slope. Since \(\frac{1}{3} \neq -6\), the lines are not parallel.
- **Perpendicular lines** have slopes that are negative reciprocals of each other. The negative reciprocal of \(\frac{1}{3}\) is \(-3\). Since \(-3 \neq -6\), the lines are not perpendicular.

Since the lines are not parallel and not perpendicular, they must be **intersecting**.

Therefore, the correct response is: **Intersecting**.