Tracking Tote/Container/Litter per Match Using API

[Strants]

Couldn't you find the per-match Tote/Auto/Litter/Coopertition/Containter/Foul from the system of equations

M_i = a_i + t_i + l_i + c_i + r_i + p_i
T_(j,t) = sum t_i P_(j,i)
T_(j,a) = sum a_i P_(j,i)
T_(j,l) = sum l_i P_(j,i)
T_(j,c) = sum c_i P_(j,i)
T_(j,r) = sum r_i P_(j,i)
T_(j,p) = sum p_i P_(j,i)

where

M_i is the score in the ith 'half-match' (since scores are kept for both red and blue alliances, we can treat a single match as two 'half-matches', one per alliance),

t_i is the tote score in the ith 'half-match',

a_i is the autonomous score in the ith 'half-match',

l_i is the litter score in the ith 'half-match',

c_i is the coopertition score in the ith 'half-match',

r_i is the recycle containter score in the ith 'half-match',

p_i is the penalty score in the ith 'half-match' (that is, the number of points lost to penalties: it may differ from -6*(number of penalties) due to the rule that match scores do not go lower than 0),

P_(j,i) is the scoring participation matrix: P_(j,i) = 1 if team j participated as a non-surrogate team in the ith 'half-match' and was not disqualified,

T_(j,*) is the total number of points team j received in all the matches they played in (note: T_(j,p) is not given directly in the rankings, but it can be calculated as (Qualification Average) * (number of matches played - # matches disqualified) - (total score from all other components). For example, the calculation for team 1619 at the Colorado Regional looks like (131.9 * 10) - (240 + 0 + 548 + 396 + 165) = -30: they probably had 5 total fouls occur in the 'half-matches' that they played).


Of course, this system might not have a unique solution (though it will always have a solution). I'd be interested if someone could give an argument that some feature of the way FIRST matches are scheduled forces the solution to be unique, or failing that at least some empirical data about how often the system is uniquely solvable.