From Documentation ("Click->Examples->Applications") we have:
At first we can use Solve:
eq = {x + 2 y + 3 z == 4, x + 4 y + 9 z == 16, x + 8 y + 27 z == 46};
Solve[eq, {x, y, z}]
(*{{x -> -5, y -> 3, z -> 1}}*)
Cramers Rule code:
cramers[m_, b_] := Module[{d = Det[m], a},Table[a = m; a[[All, k]] = b; Det[a]/d, {k, Length[m]}]]
m = {{1, 2, 3}, {1, 4, 9}, {1, 8, 27}};
b = {4, 16, 46};
x = cramers[m, {4, 16, 46}]
(*{-5, 3, 1}*)
At the end check if ok.
m.x == b
(*True*)