The problem asks for the slope of the lines determined by two points $A$ and $B$ for two cases:

1. Slope formula: The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated using the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Let's solve each part:

a. For points $A(3,4)$ and $B(4,3)$:

$m = \frac{3 - 4}{4 - 3} = \frac{-1}{1} = -1$ So, the slope is $m = -1$.

b. For points $A(4,5)$ and $B(8,13)$:

$m = \frac{13 - 5}{8 - 4} = \frac{8}{4} = 2$ So, the slope is $m = 2$.

Would you like more detailed explanations or have further questions? Here are some related questions:

1. How can the slope formula be used in real-life applications?
2. What is the significance of a positive versus a negative slope?
3. How would you find the equation of a line given its slope and a point?
4. What is the difference between a slope of zero and an undefined slope?
5. How does changing one of the points affect the slope?

Tip: When the slope is positive, the line rises from left to right, while a negative slope means it falls from left to right.