Regression for single input, multiple output

Question

I have an equation:

Y = R+B+G, where, R, B and G are the independent variables and Y is the dependent variable.

I trained a regression based model using tensorflow, that could guess the function and give value of Y for given values of R, B and G. I tried to learn the mathematics behind it implement linear regression without any API. I came across this and tried to approximate the function Y = 2X. The basic equations behind the estimation were:

    Y = βo + β1(X)+ ∈, where

    β1 = Σ(xi - xmean)(yi-ymean)/ Σ (xi - xmean)² where i= 1 to n (no. of obs.)

    βo = ymean - β1(xmean)       

I am positive that the same can be used for my problem equation: Y = R+B+G.

Soon, I realized that I wanted to do the inverse of what I had done. i.e., find R, B and G for given values of Y. Luckily, with some help, I managed to do it using Tensorflow. But, this time, when I went looking for the mathematics behind it (to implement it without any API), I could not get anything useful.

Kindly help me approximate values of R, B and G (DV) for given values of Y(IV).