The problem states:

Given the vertices of a triangle:

• $A(-1; 2)$, $B(3; -1)$, and $C(0; 4)$.
  Through each vertex, draw a line parallel to the opposite side of the triangle.

Step-by-Step Solution

We need to find the equations of the lines that pass through each vertex and are parallel to the opposite side of the triangle.

1. General equation of a straight line:
   $y - y_0 = m(x - x_0)$
   where $m$ is the slope of the line and $(x_0, y_0)$ is a point on the line.

2. Find the slopes of the triangle sides:

   • Slope of side $BC$:
     The slope between $B(3, -1)$ and $C(0, 4)$:
     $m_{BC} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-1)}{0 - 3} = \frac{5}{-3} = -\frac{5}{3}.$

   • Slope of side $AC$:
     The slope between $A(-1, 2)$ and $C(0, 4)$:
     $m_{AC} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 2}{0 - (-1)} = \frac{2}{1} = 2.$

   • Slope of side $AB$:
     The slope between $A(-1, 2)$ and $B(3, -1)$:
     $m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 2}{3 - (-1)} = \frac{-3}{4} = -\frac{3}{4}.$

3. Find the lines parallel to each side through the opposite vertex:

   • Line through $A(-1, 2)$ parallel to $BC$:
     The slope of $BC$ is $m = -\frac{5}{3}$, and the line passes through $A(-1, 2)$.
     Using the point-slope form:
     $y - 2 = -\frac{5}{3}(x - (-1)).$ Simplify:
     $y - 2 = -\frac{5}{3}(x + 1) \implies y = -\frac{5}{3}x - \frac{5}{3} + 2.$ $y = -\frac{5}{3}x + \frac{1}{3}.$

   • Line through $B(3, -1)$ parallel to $AC$:
     The slope of $AC$ is $m = 2$, and the line passes through $B(3, -1)$:
     $y - (-1) = 2(x - 3).$ Simplify:
     $y + 1 = 2(x - 3) \implies y = 2x - 6 - 1.$ $y = 2x - 7.$

   • Line through $C(0, 4)$ parallel to $AB$:
     The slope of $AB$ is $m = -\frac{3}{4}$, and the line passes through $C(0, 4)$:
     $y - 4 = -\frac{3}{4}(x - 0).$ Simplify:
     $y - 4 = -\frac{3}{4}x \implies y = -\frac{3}{4}x + 4.$

Final Equations of the Lines:

1. Line through $A(-1, 2)$ parallel to $BC$:
   $y = -\frac{5}{3}x + \frac{1}{3}.$

2. Line through $B(3, -1)$ parallel to $AC$:
   $y = 2x - 7.$

3. Line through $C(0, 4)$ parallel to $AB$:
   $y = -\frac{3}{4}x + 4.$

Do you need me to graph these lines or provide further clarifications?

Related Questions:

1. How do you find the perpendicular line to a given line through a point?
2. What are the properties of slopes of parallel and perpendicular lines?
3. How do you find the equation of a line in general form $Ax + By + C = 0$?
4. How can you determine the area of a triangle given its vertices?
5. How can you verify if three points form a straight line?

Tip: To find parallel lines, always ensure the slopes are equal. For perpendicular lines, the slopes multiply to $-1$.