The image contains a mathematical problem based on the graphs of two linear equations. The specific tasks are:

1. Equations provided: $G_1: y = -3x - 3 \quad \text{and} \quad G_2: y = \frac{1}{3}x - 2$

2. Questions:

   • (a) State the coordinates of the points where each of the graphs intersects:
     • i. the x-axis, indicated by $A$ and $B$,
     • ii. the y-axis, indicated by $C$ and $D$.
     • The two graphs intersect at the point $P$.
   • (b) Find the exact area of the triangle $ABD$.

Solution Steps:

(a) Finding the intersection points:

i. Intersection with the x-axis:

For intersection with the x-axis, $y = 0$.

• For $G_1: y = -3x - 3$: $0 = -3x - 3$ $3x = -3 \implies x = -1$ So, point $A$ is $(-1, 0)$.

• For $G_2: y = \frac{1}{3}x - 2$: $0 = \frac{1}{3}x - 2$ $\frac{1}{3}x = 2 \implies x = 6$ So, point $B$ is $(6, 0)$.

ii. Intersection with the y-axis:

For intersection with the y-axis, $x = 0$.

• For $G_1: y = -3x - 3$: $y = -3(0) - 3 = -3$ So, point $C$ is $(0, -3)$.

• For $G_2: y = \frac{1}{3}x - 2$: $y = \frac{1}{3}(0) - 2 = -2$ So, point $D$ is $(0, -2)$.

(b) Finding the exact area of the triangle $ABD$:

The vertices of the triangle $ABD$ are $A(-1, 0)$, $B(6, 0)$, and $D(0, -2)$.

Using the formula for the area of a triangle given three vertices $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$: $\text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|$ Substituting the points $A(-1, 0)$, $B(6, 0)$, and $D(0, -2)$: $\text{Area} = \frac{1}{2} \left| -1(0 - (-2)) + 6(-2 - 0) + 0(0 - 0) \right|$ $= \frac{1}{2} \left| -1(2) + 6(-2) + 0 \right|$ $= \frac{1}{2} \left| -2 - 12 \right|$ $= \frac{1}{2} \times 14 = 7 \, \text{square units}$

Final Answers:

1. Points of Intersection:

   • $A(-1, 0)$
   • $B(6, 0)$
   • $C(0, -3)$
   • $D(0, -2)$

2. Area of Triangle $ABD$: $7$ square units.

Would you like more details on any of these calculations or explanations?

Here are some related questions to consider:

1. What is the general process for finding the intersection of two lines?
2. How does changing the slope of a line affect its intersection points with the axes?
3. Can you determine the area of a triangle formed by any three points on a graph?
4. How would the area calculation change if the coordinates of point $D$ were different?
5. What are other methods to find the area of a triangle if you know the vertices?

Tip: When finding where a graph intersects the x-axis or y-axis, remember that the value of the other coordinate (y or x, respectively) is zero.