Let's go through these questions one by one and solve them.

#2:

Write a linear equation in point-slope form of the line that passes through the point $(2, -4)$ and is perpendicular to $y = \frac{2}{5}x - 4$.

• Step 1: Find the slope of the given line. The slope of $y = \frac{2}{5}x - 4$ is $m = \frac{2}{5}$.
• Step 2: For perpendicular lines, the slopes are negative reciprocals. The slope of the perpendicular line is $m_{\perp} = -\frac{5}{2}$.
• Step 3: Use the point-slope formula $y - y_1 = m(x - x_1)$ with the point $(2, -4)$ and slope $-\frac{5}{2}$. $y + 4 = -\frac{5}{2}(x - 2)$

#3:

Write a linear equation in point-slope form of the line that passes through the point $(-2, 3)$ and is parallel to $2x + y = -5$.

• Step 1: Rewrite the given equation in slope-intercept form. $2x + y = -5 \implies y = -2x - 5$ The slope is $m = -2$.
• Step 2: For parallel lines, the slopes are the same. So the slope of the new line is $m = -2$.
• Step 3: Use the point-slope form with the point $(-2, 3)$ and slope $-2$. $y - 3 = -2(x + 2)$

#4:

Write a linear equation in standard form of the line that passes through the point $(2, 0)$ and is perpendicular to $y = \frac{1}{2}x$.

• Step 1: The slope of the given line is $m = \frac{1}{2}$.
• Step 2: The slope of the perpendicular line is $m_{\perp} = -2$.
• Step 3: Use point-slope form to find the equation. $y - 0 = -2(x - 2) \implies y = -2x + 4$
• Step 4: Convert to standard form. $2x + y = 4$

#5:

Write a linear equation in point-slope form of the line that passes through the point $(-2, -1)$ and is perpendicular to $2x - y = -2$.

• Step 1: Rewrite the given equation in slope-intercept form. $2x - y = -2 \implies y = 2x + 2$ The slope is $m = 2$.
• Step 2: The slope of the perpendicular line is $m_{\perp} = -\frac{1}{2}$.
• Step 3: Use the point-slope form with the point $(-2, -1)$ and slope $-\frac{1}{2}$. $y + 1 = -\frac{1}{2}(x + 2)$

#6:

Write a linear equation in slope-intercept form of the line that passes through the point $(3, 0)$ and is perpendicular to $y = 4$.

• Step 1: The line $y = 4$ is a horizontal line with slope $m = 0$.
• **Step