Using ML in solving 'm' variables and constants, and 'n' equations to get the ranking factor constants k1, k2, k3...

[R. Rajesh Jeba Anbiah]

How to find weightage constants from weightage factors and ranks?

There are couple of weightage factors in deciding ranks (say, in music or art or competition). For some factors, higher value is better and for others lower value is better.

The committee is publishing the weightage factors, weightage values and final rank--not the weightage constants.

For example:

Person1 - Rank: 1 

 Factor1 - 58 (lower is better) x Weightage constant K1 - Unknown (?) 

 Factor2 - 8 (lower is better) x Weightage constant K2 - Unknown (?) 

 Factor3 - 238 (higher is better) x Weightage constant K3 - Unknown (?) 

 Factor4 - 15336 (higher is better) x Weightage constant K4 - Unknown (?) 

 Factor n - .... 

Meaning/IOW, (58 x k1) + (8 x k2) + (238 x k3) + (15536 x k4) + (... x kn) = 1

Person2 - Rank: 2 

 Factor1 - 28 (lower is better) x Weightage constant K1 - Unknown (?) 

 Factor2 - 2 (lower is better) x Weightage constant K2 - Unknown (?) 

 Factor3 - 5555 (higher is better) x Weightage constant K3 - Unknown (?) 

 Factor4 - 45336 (higher is better) x Weightage constant K4 - Unknown (?) 

 Factor n - .... 

Meaning/IOW, (28 x k1) + (2 x k2) + (5555 x k3) + (45336 x k4) + (... x kn) = 2

In a formula representation:

 f11 x k1 + f12 x k2 + f13 x k3... + f1m x km = 1 (Rank) 

 f21 x k1 + f22 x k2 + f23 x k3... + f2m x km = 2 

 ... 

 fn1 x k1 + fn2 x k2 + fn3 x k3... + fnm x km = n

Or, at least, the values in reverse order, as our aim to get the Rank

  f11 x k1 + f12 x k2 + f13 x k3... + f1m x km = n (some higher value for the Rank) 

 f21 x k1 + f22 x k2 + f23 x k3... + f2m x km = .. 

 ... 

 fn1 x k1 + fn2 x k2 + fn3 x k3... + fnm x km = 1

Here, we need to solve/find the constants k1, k2...

I was trying to solve these 'n' equations with 'm' variables 'm' constants, to get the ranking factor constants k1, k2, k3... km. 

Recently, I came across some basic ML theories that suggest that it must be possible to solve formula like these using ML. (If I understand right, ML is all about solving these type of equations) Can someone please direct me to the appropriate library or solution for solving this? TIA