I have for example simple neural network - one neuron and 2 variables.
For example we have binary AND action(0 AND 0 = 0 AND 1 = 1 AND 0 = 0, 1 AND 1 = 1). So we need to find such weights of w1*x + w2*y = T( or y = k*x + T, where k = -w1/w2) to neural network can work.
Graphically it look's like:
So as you can see T - is not any random value. If T = 0.5 there are no such w1 and w2 to make it work (you can't draw line y = kx + T to separate 1;1 from other points). To learn my neural network I use simple learning with teacher.