\(M\_HC\ is\ our\ matrix\ that\ maps\ the\ first\ three\ hydrocarbons\ to\ number\ of\ Hydrogens\ and\ Carbons\)

(Subscript[M, HC])  =  {
{4, 1}, 
{8, 
            3}, 
{10, 
            4}
};

    
(Subscript[M, HC]) // MatrixForm

PTmatrix  =  {
{α, β}, 
{γ, δ}, 
{ϵ, ϕ}
};
PTmatrix // MatrixForm

MPT   =  (Subscript[M, HC]) .   PTmatrix

Clear[MPT]

MPT  =  
        Transpose[(Subscript[M, HC])] .   PTmatrix;
MPT // MatrixForm

MPTinverse  = Factor[Inverse[MPT]];
MPTinverse // MatrixForm

Det[MPT]

RowReduce[(Subscript[M, HC])]

Same example for water and water complexes

watmat  =  {{2, 4}, {1, 2}};
watmat // MatrixForm

molvec  =  {h20, h402}

atomvec  =  {h, o}

eqs[[1]]

atomvec // MatrixForm

eq[1]  =  (watmat . molvec)[[1]]  ==  
        atomvec[[1]]

eq[2]  =   (watmat . molvec)[[2]]  ==  
        atomvec[[2]]

Solve[{eq[1], eq[2]}, molvec]

Eliminate[{eq[1], eq[2]}, molvec]