Let's say we want to predict the following neural network's output prediction when it has been trained with these weights:

We will call the first 3 nodes in the first layer A, B, and C. We will call the two hidden nodes D and E. With these node names, we get the following equation for y (we are setting the bias to 0 for simplicity):

y = sigmoid(4D + 2E)

To get the equation for y in terms of the input nodes, we must write out the equation for nodes D and E:

The equation for y in terms of the input nodes is obtained by plugging in these equations for D and E back into the equation for y (again, remember that we are setting the bias to 0 for simplicity):