Let's solve the given problems step by step.

Given Points:

• $A(-2, 3)$
• $B(4, 5)$
• $C(-2, 4)$

(a) Equation of the line passing through points $A$ and $B$

To find the equation of the line passing through two points, we use the formula for the slope $m$: $m = \frac{y_2 - y_1}{x_2 - x_1}$

For points $A(-2, 3)$ and $B(4, 5)$: $m = \frac{5 - 3}{4 - (-2)} = \frac{2}{6} = \frac{1}{3}$

Using the point-slope form of the line equation: $y - y_1 = m(x - x_1)$

Using point $A(-2, 3)$: $y - 3 = \frac{1}{3}(x - (-2))$ $y - 3 = \frac{1}{3}(x + 2)$ $y - 3 = \frac{1}{3}x + \frac{2}{3}$ $y = \frac{1}{3}x + \frac{2}{3} + 3$ $y = \frac{1}{3}x + \frac{2}{3} + \frac{9}{3}$ $y = \frac{1}{3}x + \frac{11}{3}$

So, the equation of the line passing through points $A$ and $B$ is: $y = \frac{1}{3}x + \frac{11}{3}$

(b) Equation of the line passing through points $A$ and $C$

For points $A(-2, 3)$ and $C(-2, 4)$:

Since both points have the same x-coordinate, the line is vertical: $x = -2$

(c) Equation of the line passing through points $B$ and $C$

For points $B(4, 5)$ and $C(-2, 4)$: $m = \frac{4 - 5}{-2 - 4} = \frac{-1}{-6} = \frac{1}{6}$

Using the point-slope form: $y - y_1 = m(x - x_1)$

Using point $B(4, 5)$: $y - 5 = \frac{1}{6}(x - 4)$ $y - 5 = \frac{1}{6}x - \frac{4}{6}$ $y - 5 = \frac{1}{6}x - \frac{2}{3}$ $y = \frac{1}{6}x - \frac{2}{3} + 5$ $y = \frac{1}{6}x - \frac{2}{3} + \frac{15}{3}$ $y = \frac{1}{6}x - \frac{2}{3} + 5$ $y = \frac{1}{6}x + \frac{13}{3}$

So, the equation of the line passing through points $B$ and $C$ is: $y = \frac{1}{6}x + \frac{13}{3}$

(d) Graph all three equations on one graph

To plot these equations, we need a coordinate system. Here are the equations we will graph:

1. $y = \frac{1}{3}x + \frac{11}{3}$
2. $x = -2$
3. $y = \frac{1}{6}x + \frac{13}{3}$

I can help you visualize this using a graph plotting tool.

Would you like me to generate the graph for you?