The problem asks us to determine whether pairs of lines are parallel, perpendicular, or neither. We need to analyze the slopes of the lines. Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals of each other (their product is -1).

1. Problem Description

The problem asks us to determine whether pairs of lines are parallel, perpendicular, or neither. We need to analyze the slopes of the lines. Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals of each other (their product is -1).

2. Solution Steps

Problem 7:

The given lines are $y = \frac{1}{4}x - 3$ and $y = -4x + 3$.

The slopes are $m_1 = \frac{1}{4}$ and $m_2 = -4$.

The product of the slopes is $m_1 \cdot m_2 = \frac{1}{4} \cdot (-4) = -1$.

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

Problem 8:

The given lines are $y = 2x - 4$ and $y = -2x + 5$.

The slopes are $m_1 = 2$ and $m_2 = -2$.

The slopes are not equal, so the lines are not parallel.

The product of the slopes is $m_1 \cdot m_2 = 2 \cdot (-2) = -4$.

Since the product of the slopes is not -1, the lines are not perpendicular.

Thus, the lines are neither parallel nor perpendicular.

Problem 9:

The given lines are $3x + y = 5$ and $y = -\frac{1}{3}x + 2$.

We need to rewrite the first equation in slope-intercept form ($y = mx + b$).

$3x + y = 5$ can be rewritten as $y = -3x + 5$.

The slopes are $m_1 = -3$ and $m_2 = -\frac{1}{3}$.

The slopes are not equal, so the lines are not parallel.

The product of the slopes is $m_1 \cdot m_2 = (-3) \cdot (-\frac{1}{3}) = 1$.

Since the product of the slopes is not -1, the lines are not perpendicular.

Thus, the lines are neither parallel nor perpendicular.

Problem 10:

The given equation is $2x + 3x - 6 = 0$. It seems there is a missing 'y=' before $2x + 3x - 6 = 0$. Also, the second equation is cut off. I cannot solve this problem with the information given.

3. Final Answer

7. Perpendicular

8. Neither

9. Neither

1

0. Cannot be determined with the given information.