Equation of a Multi-Layer Perceptron Network
You are forgetting one element of the MLP which is the activation function. If your activation function is linear - then you can simply flatten out all the neurons into one single linear equation. The advantage of MLP however is its non-linearities so I suspect in your network you do have some activation (sigmoid? tanh? relu? etc..).
As for your graph - you could simply output predictions from your MLP and plot the exact scatter plot you have above. The only difference would be you wouldn't have a simple way of expressing this network in algebraic notation (as you have done on the existing x-axis).
To describe networks effectively in text you should look into matrix notation describing the weights and inputs of each layer. Maybe take a look at something like this to get started: https://www.jeremyjordan.me/intro-to-neural-networks/