Clear[t];
Feed[0] = Null;
Feed[eqn_] := Module[{i, eqn1},
i = Min[Cases[eqn, x[k_] :> k, Infinity]];
eqn1 = Expand[eqn/ Coefficient[eqn, x[i], 1]];
If[Head[t[i]] =!= t,
(*then*) Feed[Simplify[t[i] - eqn1]],
(*else*) t[i] = Simplify[eqn1]
];
];

n=5;
MatrixForm[M=Table[RandomInteger[100], {n}, {n}]]

eqns=(M-z IdentityMatrix[n]).Array[x, {n}]

(Feed[#]; Table[t[i], {i,n}])& /@ eqns

Feed[eqns[[1]]]

Table[t[i], {i,4}]

Feed[eqns[[2]]]

Table[t[i], {i,4}]

Feed[eqns[[3]]];
Table[t[i], {i,4}]

Feed[eqns[[4]]];
Table[t[i], {i,4}]