Problem 1 (Point-Slope Form): Find the equation of the line passing through the point $P(-2, 5)$ with a slope of $3$. Write the final equation in standard general form ($Ax + By + C = 0$).

Problem 2 (Slope-Intercept Form): A line has a y-intercept of $-4$ and a slope of $-\frac{1}{2}$. Find its equation in general form.

Problem 3 (Two-Point Form): Determine the equation of the line passing through $A(3, -1)$ and $B(6, 5)$.

Problem 4 (Intercept Form): A line has an x-intercept of $5$ and a y-intercept of $3$. What is its equation?

Problem 1: The perpendicular distance from the origin to a line is $4$ units, and the angle this perpendicular makes with the positive x-axis is $60^\circ$. Find the equation of the line.

Problem 2: Convert the general equation $3x - 4y + 10 = 0$ into normal form and find the distance to the origin.

Problem 3: Write the equation $x + y - 6 = 0$ in normal form. What is the angle of the normal?

Case Study 1: Analyze the family of lines given by the equation $(2k+1)x + (k-3)y + (5-4k) = 0$. Prove that all lines in this family pass through a single fixed point, regardless of the value of $k$, and find that point.

Case Study 2: A civil engineer is designing a series of perfectly parallel drainage pipes. The master blueprint line is given by $3x - 4y + 12 = 0$. Express the mathematical family representing all possible parallel pipes, and determine the specific pipe equation that passes exactly through the survey point $(5, -2)$.

Problem 1: Find the equation of a line passing through $(4, 1)$ that is parallel to the line $2x - 5y = 10$.

Problem 2: Determine the equation of the line passing through $(-3, 2)$ that is strictly perpendicular to $x + 3y + 6 = 0$.

Problem 3: Are the lines $4x - 6y = 9$ and $3x + 2y = 5$ parallel, perpendicular, or neither?

Problem 1: Calculate the precise perpendicular distance from the point $(3, -2)$ to the line $5x - 12y + 4 = 0$.

Problem 2: Find the altitude of a triangle drawn from vertex $A(-1, 4)$ to the side formed by the line passing through $B(2, -3)$ and $C(5, 1)$.

Problem 1: Calculate the exact distance separating the two parallel lines $3x + 4y - 10 = 0$ and $3x + 4y + 5 = 0$.

Problem 2: Find the separation distance between the parallel lines $8x - 6y + 7 = 0$ and $4x - 3y - 2 = 0$.

Problem 1: Find the acute angle between the intersecting lines $2x - y + 5 = 0$ and $3x + y - 4 = 0$.

Problem 2: What is the angle between the lines $y = \frac{1}{2}x + 3$ and $y = 3x - 1$?