The force exerted by the wedge is equal to
You're actually dealing with an equilibrium problem. However, the trick here is to distinguish between translational equilibrium and rotational equilibrium.
For example, let's assume that the first thing you want to determine is whether or not the plank is in rotational equilibrium.
In this case, equilibrium will be established if the counterclockwise torque caused by the boy is equal to the clockwise torque caused by the girl.
Torque is simply a term used to describe a force's tendency to induce rotation and is equal to
So, in order to be at equilibrium, you need to have
Therefore, the force exerted by the fulcrum would be zero, right? Wrong. This is where translational equilibrium comes into play.
In order for a body to be at equilibrium, the vector sum of all the forces that are acting on that body must be equal to zero.
This implies that the two forces that push downward, i.e. the weights of the children, must be equal to the force pushing upward, i.e. the reactive force exerted by the fulcrum.
This implies that
I think the minus sign is optional, since it just signifies that this force is pointed upwards, as opposed to the other two forces which point downwards.