To find the equation of the line $p: y = kx + q$ passing through the points $A(-1, 3)$ and $B(2, 6)$, we need to follow these steps:

1. Calculate the slope $k$

The slope $k$ (also called the gradient) is calculated using the formula:

$k = \frac{y_2 - y_1}{x_2 - x_1}$

Substitute the coordinates of $A(x_1 = -1, y_1 = 3)$ and $B(x_2 = 2, y_2 = 6)$:

$k = \frac{6 - 3}{2 - (-1)} = \frac{3}{3} = 1$

2. Find the intercept $q$

Now that we know $k = 1$, we can use one of the points (say $A$) to find the intercept $q$. Using the equation $y = kx + q$, substitute $x = -1$, $y = 3$, and $k = 1$:

$3 = 1(-1) + q$

$3 = -1 + q \quad \Rightarrow \quad q = 4$

Final Equation

Thus, the equation of the line is:

$y = x + 4$

Would you like further details or have any questions? Here are some related questions you might find useful:

1. How can we determine if two lines are parallel?
2. What does the slope of a line represent geometrically?
3. How do you find the equation of a line given its slope and a point on the line?
4. How can you check if a point lies on a given line?
5. What happens when two points are vertically or horizontally aligned?

Tip: The slope of a line is a measure of how steep the line is—positive slopes rise from left to right, while negative slopes fall.