Question

Determine whether the two lines are​ parallel, perpendicular, or neither parallel nor perpendicular.
3 x minus 2 y equals 16
2 x plus 3 y equals 7
Question content area bottom
Part 1
A.
Parallel
B.
Perpendicular
C.
Neither

Answer 1

To determine whether the two lines are parallel, perpendicular, or neither, we need to find the slopes of both lines from their equations.

**First line: \(3x - 2y = 16\)**

1. Rearranging into slope-intercept form \(y = mx + b\):
\[
-2y = -3x + 16
\]
\[
y = \frac{3}{2}x - 8
\]
The slope \(m_1\) of the first line is \(\frac{3}{2}\).

**Second line: \(2x + 3y = 7\)**

2. Rearranging into slope-intercept form:
\[
3y = -2x + 7
\]
\[
y = -\frac{2}{3}x + \frac{7}{3}
\]
The slope \(m_2\) of the second line is \(-\frac{2}{3}\).

**Determine the relationship between the slopes:**

- Two lines are **parallel** if their slopes are equal: \(m_1 = m_2\).
- Two lines are **perpendicular** if the product of their slopes is \(-1\): \(m_1 \cdot m_2 = -1\).

Calculating the product of the slopes:
\[
m_1 \cdot m_2 = \frac{3}{2} \cdot \left(-\frac{2}{3}\right) = -1
\]

Since the product of the slopes is \(-1\), the lines are perpendicular.

**Answer: B. Perpendicular**