payments={{2000,1800,1500,1600},{2200,1600,1400,1400},{2500,2000,1700,1600}};
%//MatrixForm

carType = Table[(Subscript[c, i]),{i,3}]
age = Table[(Subscript[a, j]),{j,4}]
(Subscript[c, 1])=1;
equations1=Table[Total[carType[[i]] age] == Total[payments[[i,;;]]],{i,2,3}]
equations2 = Table[Total[carType age[[j]]] == Total[payments[[;;,j]]],{j,4}]
equations = Join[equations1,equations2]

initCond = Join[Map[{#,1}&,carType[[2;;]]],Map[{#,1}&,age]]

solution=FindRoot[equations,initCond]

age=age/.solution
carType = carType/.solution

tariffs=Outer[Times,age,carType]
%//TableForm
payments//TableForm

Total[Flatten[payments]]
Total[Flatten[tariffs]]