From the graph, find the gradient and equation of each of the lines through points A and B, B and C, and A and C. Hence, determine the x and y-intercepts.

Step 1: Identify the coordinates of the points from the graph.

• Point A: $(0, 3)$
• Point B: $(4, 3)$
• Point C: $(4, 0)$

a) Find the gradient of each of the lines through points A and B, and through points B and C.
The formula for the gradient $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $m = \frac{y_2 - y_1}{x_2 - x_1}$.

For line AB:
Points A$(0, 3)$ and B$(4, 3)$.

$$m_{AB} = \frac{3 - 3}{4 - 0} = \frac{0}{4} = 0$$

The gradient of line AB is $\boxed{0}$.

For line BC:
Points B$(4, 3)$ and C$(4, 0)$.

$$m_{BC} = \frac{0 - 3}{4 - 4} = \frac{-3}{0}$$

Division by zero means the gradient is undefined. This is a vertical line.
The gradient of line BC is $\boxed{undefined}$.

b) Find the equation of each of the lines through points A and B, and through points B and C.
The formula for the equation of a line is $y - y_1 = m(x - x_1)$.

For line AB:
Gradient $m_{AB} = 0$. Using point A$(0, 3)$.

$$y - 3 = 0(x - 0)$$ $$y - 3 = 0$$ $$y = 3$$

The equation of line AB is $\boxed{y = 3}$.

For line BC:
The line BC is a vertical line passing through $x=4$.
The equation of a vertical line is $x = k$, where $k$ is the x-coordinate.
The equation of line BC is $\boxed{x = 4}$.

c) Find the gradient and equation of the line passing through points A and C. Hence, determine x and y-intercepts.

For line AC:
Points A$(0, 3)$ and C$(4, 0)$.

Gradient $m_{AC}$:

$$m_{AC} = \frac{0 - 3}{4 - 0} = \frac{-3}{4}$$

The gradient of line AC is $\boxed*{-\frac{3}{4}*}$.

Equation of line AC:
Using point A$(0, 3)$ and $m_{AC} = -\frac{3}{4}$.

$$y - 3 = -\frac{3}{4}(x - 0)$$ $$y - 3 = -\frac{3}{4}x$$ $$y = -\frac{3}{4}x + 3$$

The equation of line AC is $\boxed*{y = -\frac{3}{4}x + 3*}$.

Determine x and y-intercepts for line AC:
To find the y-intercept, set $x = 0$:

$$y = -\frac{3}{4}(0) + 3$$ $$y = 3$$

The y-intercept is $\boxed*{(0, 3)}*$.

To find the x-intercept, set $y = 0$:

$$0 = -\frac{3}{4}x + 3$$ $$\frac{3}{4}x = 3$$ $$3x = 12$$ $$x = 4$$

The x-intercept is $\boxed*{(4, 0)}*$.