A taxi company charges a fixed booking fee plus a cost that varies directly with the distance travelled. Lubuto pays K45.50 for a 10km trip and K65.00 for a 20km trip. (a) Write an equation for Lubuto's fare (b) Determine the fixed booking fee (c) How much will a 35km trip cost Lubuto?

Let $F$ be the total fare in K, and $d$ be the distance travelled in km.
Let $C$ be the fixed booking fee in K, and $k$ be the cost per km in K/km.
The problem states that the fare consists of a fixed booking fee plus a cost that varies directly with the distance travelled. This can be expressed as:

We are given two scenarios:

1. For a 10 km trip, the fare is K45.00: $$45 = C + 10k \quad (1)$$
2. For a 20 km trip, the fare is K65.00: $$65 = C + 20k \quad (2)$$

Subtract equation (1) from equation (2):

Now substitute $k=2$ into equation (1):

So, the fixed booking fee is K25.00 and the cost per km is K2.00.

Substitute the values of $C$ and $k$ into the general fare equation $F = C + kd$.

The equation for Lubuto's fare is $\boxed{F = 25 + 2d}$.

From Step 2, we found the value of $C$.
The fixed booking fee is $\boxed{\text{K}25.00}$.

Use the equation found in part (a), $F = 25 + 2d$, and substitute $d=35$ km.

A 35 km trip will cost $\boxed{\text{K}95.00}$.