To determine which of the given lines are perpendicular to each other, we need to examine their slopes. Lines are perpendicular if the product of their slopes is $-1$.

First, we rearrange each equation into the slope-intercept form $y = mx + b$, where $m$ is the slope.

Line (a):

$-3x + 4y + 4 = 0$ Rearrange to solve for $y$: $4y = 3x - 4$ $y = \frac{3}{4}x - 1$ Slope $m_a = \frac{3}{4}$

Line (b):

$-4x - 3y = 0$ Rearrange to solve for $y$: $-3y = 4x$ $y = -\frac{4}{3}x$ Slope $m_b = -\frac{4}{3}$

Line (c):

$-3x - 4y = 3$ Rearrange to solve for $y$: $-4y = 3x - 3$ $y = -\frac{3}{4}x + \frac{3}{4}$ Slope $m_c = -\frac{3}{4}$

Line (d):

$4x + 3y + 5 = 0$ Rearrange to solve for $y$: $3y = -4x - 5$ $y = -\frac{4}{3}x - \frac{5}{3}$ Slope $m_d = -\frac{4}{3}$

Line (e):

$3x - 4y + 9 = 0$ Rearrange to solve for $y$: $-4y = -3x - 9$ $y = \frac{3}{4}x + \frac{9}{4}$ Slope $m_e = \frac{3}{4}$

Line (f):

$-4x + 3y - 4 = 0$ Rearrange to solve for $y$: $3y = 4x + 4$ $y = \frac{4}{3}x + \frac{4}{3}$ Slope $m_f = \frac{4}{3}$

Checking for Perpendicularity

Lines are perpendicular if the product of their slopes is $-1$. Let’s check the products of the slopes:

• $m_a \times m_b = \left(\frac{3}{4}\right) \times \left(-\frac{4}{3}\right) = -1$
• $m_a \times m_d = \left(\frac{3}{4}\right) \times \left(-\frac{4}{3}\right) = -1$
• $m_a \times m_f = \left(\frac{3}{4}\right) \times \left(\frac{4}{3}\right) = 1$ (not perpendicular)
• $m_c \times m_f = \left(-\frac{3}{4}\right) \times \left(\frac{4}{3}\right) = -1$
• $m_e \times m_b = \left(\frac{3}{4}\right) \times \left(-\frac{4}{3}\right) = -1$
• $m_e \times m_d = \left(\frac{3}{4}\right) \times \left(-\frac{4}{3}\right) = -1$

Therefore, the lines that are perpendicular to each other are:

• (a) and (b)
• (a) and (d)
• (c) and (f)
• (e) and (b)
• (e) and (d)

Would you like any more details or have any questions? Here are some related questions:

1. How do you find the slope of a line given its equation in standard form?
2. Can two lines with slopes of $\frac{2}{3}$ and $-\frac{2}{3}$ be perpendicular?
3. What is the slope of a line perpendicular to $y = 5x - 7$?
4. If two lines are parallel, what can be said about their slopes?
5. How do you find the equation of a line perpendicular to a given line and passing through a specific point?

Tip: Always simplify the equation of a line to the slope-intercept form $y = mx + b$ to easily identify the slope.