To find the equation of a line that passes through a given point and is perpendicular to a given line, we follow these steps:

1. Identify the slope of the given line.
2. Find the perpendicular slope, which is the negative reciprocal of the slope of the given line.
3. Use the point-slope form of a line equation: $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is the given point.

Let's solve each problem step by step:

4. $y = -2x - 1$; Point: $(2, -1)$

• The slope of the given line $y = -2x - 1$ is $m = -2$.
• The slope of the perpendicular line is the negative reciprocal of $-2$, which is $\frac{1}{2}$.
• Using the point-slope form: $y - (-1) = \frac{1}{2}(x - 2)$ Simplify: $y + 1 = \frac{1}{2}(x - 2)$ Distribute the $\frac{1}{2}$: $y + 1 = \frac{1}{2}x - 1$ Subtract 1 from both sides: $y = \frac{1}{2}x - 2$
• Equation of the perpendicular line: $y = \frac{1}{2}x - 2$.

5. $y + 4 = -\frac{2}{3}(x - 2)$; Point: $(4, -2)$

• First, identify the slope of the given line: The equation is already in point-slope form. The slope is $m = -\frac{2}{3}$.
• The slope of the perpendicular line is the negative reciprocal of $-\frac{2}{3}$, which is $\frac{3}{2}$.
• Using the point-slope form: $y - (-2) = \frac{3}{2}(x - 4)$ Simplify: $y + 2 = \frac{3}{2}(x - 4)$ Distribute the $\frac{3}{2}$: $y + 2 = \frac{3}{2}x - 6$ Subtract 2 from both sides: $y = \frac{3}{2}x - 8$
• Equation of the perpendicular line: $y = \frac{3}{2}x - 8$.

6. $x - 6y = -2$; Point: $(-5, 6)$

• First, rewrite the given equation in slope-intercept form $y = mx + b$: $x - 6y = -2$ Solve for $y$: $-6y = -x - 2$ Divide by -6: $y = \frac{1}{6}x + \frac{1}{3}$
• The slope of the given line is $m = \frac{1}{6}$.
• The slope of the perpendicular line is the negative reciprocal of $\frac{1}{6}$, which is $-6$.
• Using the point-slope form: $y - 6 = -6(x - (-5))$ Simplify: $y - 6 = -6(x + 5)$ Distribute the $-6$: $y - 6 = -6x - 30$ Add 6 to both sides: $y = -6x - 24$
• Equation of the perpendicular line: $y = -6x - 24$.

---

If you have any questions or need further details, feel free to ask!

Here are 5 related questions:

1. What is the formula for finding the slope of a line from its equation?
2. How do you derive the negative reciprocal of a given slope?
3. What is the geometric interpretation of perpendicular lines on a graph?
4. Can two perpendicular lines have the same slope?
5. How do you convert a line equation from standard form to slope-intercept form?

Tip: The negative reciprocal of a slope $m$ is $-\frac{1}{m}$. Always use this to find the perpendicular slope!