We are given two equations and asked to determine if their corresponding lines are parallel, perpendicular, or neither.

The equations are:

1. $-4x = 2y + 5$
2. $-4y = 8x + 3$

Step 1: Rewrite the equations in slope-intercept form ($y = mx + b$).

First equation: $-4x = 2y + 5$

Solve for $y$:

$-4x - 5 = 2y$ $y = \frac{-4x - 5}{2}$ $y = -2x - \frac{5}{2}$

The slope ($m_1$) of the first line is $-2$.

Second equation: $-4y = 8x + 3$

Solve for $y$:

$y = \frac{8x + 3}{-4}$ $y = -2x - \frac{3}{4}$

The slope ($m_2$) of the second line is also $-2$.

Step 2: Compare the slopes.

• If the slopes are equal, the lines are parallel.
• If the product of the slopes is $-1$, the lines are perpendicular.
• If neither of these conditions is true, the lines are neither.

Here, both slopes are $-2$, so the lines are parallel.

Would you like more details, or do you have any questions?

Here are 5 related questions to consider:

1. How do you calculate the y-intercept from an equation in slope-intercept form?
2. What happens when two lines have the same slope but different y-intercepts?
3. How can you verify graphically that two lines are parallel?
4. What method can be used to find the angle between two intersecting lines?
5. Can two lines be neither parallel nor perpendicular?

Tip: Always rearrange equations into slope-intercept form to easily determine their slope.