Draw the graph of 3y = 9x Show all calculations. Find the equation of this graph.

We know that $729 = 3^6$.
Therefore, $\frac{1}{729} = \frac{1}{3^6} = 3^{-6}$.
The equation becomes:

The value of $x$ is $\boxed{\text{4.5}}$.

Question 5.1
Given pattern: $15; 12; 9; 6; 3; \dots$

5.1.1: Describe the pattern in words.
The pattern is an arithmetic sequence where each term is obtained by subtracting 3 from the previous term. The common difference is $-3$.

5.1.2: Write the next three terms of the pattern given above.
The last given term is 3.

• Next term: $3 - 3 = 0$
• Next term: $0 - 3 = -3$
• Next term: $-3 - 3 = -6$ The next three terms are $\boxed{\text{0, -3, -6}}$.

5.1.3: Determine the $n^{th}$ term of this pattern.

The first term $a = 15$.
The common difference $d = 12 - 15 = -3$.

The $n^{th}$ term is $\boxed{T_n = 18 - 3n}$.

5.1.4: Determine the position of the term with the value of $-45$.

The position of the term with the value of $-45$ is $\boxed{\text{21}}$.

5.1.5: What will be the $50^{th}$ term?

The $50^{th}$ term will be $\boxed{\text{-132}}$.

Question 6.1: Draw the graph of $3y = 9x - 18$. Show all calculations.

Divide all terms by 3:

The y-intercept is the point where $x=0$.

The y-intercept is $(0, -6)$.

The x-intercept is the point where $y=0$.

The x-intercept is $(2, 0)$.

Then, draw a straight line passing through these two points.

Question 6.2: Find the equation of this graph.

From the graph, we can see the line passes through:

• The y-intercept: $(0, -2)$
• The x-intercept: $(-2, 0)$

The formula for slope is $m = \frac{y_2 - y_1}{x_2 - x_1}$.

We found the slope $m = -1$.
The y-intercept $c$ is the y-coordinate where the line crosses the y-axis, which is $-2$.
Substitute $m = -1$ and $c = -2$ into the equation:

The equation of the graph is $\boxed{y = -x - 2}$.

Question 7.1.1: Calculate the size of $x$ on the given diagram below.

The angles given are $(x+30^\circ)$, $(x+40^\circ)$, and $(2x+10^\circ)$.

The size of $x$ is $\boxed{25^\circ}$.

Question 7.1.2: Determine the size of $p$, $q$ and $r$.

Given that lines $KL$ and $MN$ are parallel, and $GH$ is a transversal.

Angle $q$ and the angle $133^\circ$ are vertically opposite angles. Vertically opposite angles are equal.

The angle adjacent to $133^\circ$ on the straight line $MN$ is $180^\circ - 133^\circ = 47^\circ$.
Angle $p$ and this $47^\circ$ angle are alternate interior angles because $KL \parallel MN$. Alternate interior angles are equal.

Alternatively, angle $p$ and the angle corresponding to $133^\circ$ (which is $133^\circ$) are supplementary angles on a straight line.

Angle $r$ and angle $p$ are vertically opposite angles. Vertically opposite angles are equal.

The sizes are $\boxed{p = 47^\circ, q = 133^\circ, r = 47^\circ}$.