Solve[{3x+2y-8z==1,-4x+y+3z==-2,6x+4y+9z==3},{x,y,z}]

N[%]

A = {{3,2,-8,1}, {-4, 1, 3, -2},{6, 4,9 ,3}};

A//MatrixForm

RowReduce[%]

%//MatrixForm

N[%]//MatrixForm

A//MatrixForm

E0 is a row swapping elementary matrix

E0 = {{0,1,0},{1,0,0},{0,0,1}};

E0.A//MatrixForm

Let us now RREF A by left multiplication of elementary matrices

A//MatrixForm

E1 = {{1/3,0,0},{0,1,0},{0,0,1}};

E1.A//MatrixForm

E2 = {{1,0,0},{4,1,0},{-6,0,1}};

E2//MatrixForm

E2.E1.A//MatrixForm

E2i = {{1,0,0},{-4,1,0},{6,0,1}};

E2i.E2//MatrixForm

E2.E2i//MatrixForm

E1//MatrixForm

E1i = {{3,0,0},{0,1,0},{0,0,1}};

E1i.E1//MatrixForm

E2.E1.A//MatrixForm

E3 = {{1,0,0},{0,3/11,0},{0,0,1}};

E3.E2.E1.A//MatrixForm

E4 = {{1,-2/3,0},{0,1,0},{0,0,1}};

E4//MatrixForm

E4.E3.E2.E1.A//MatrixForm

E5 = {{1,0,0},{0,1,0},{0,0,1/25}};

E5.E4.E3.E2.E1.A//MatrixForm

E6 = {{1,0,14/11},{0,1,23/11},{0,0,1}};

E6//MatrixForm

E6.E5.E4.E3.E2.E1.A//MatrixForm

RowReduce[A]//MatrixForm

A1 = A[[All,{1,2,3}]]

A1//MatrixForm

X = {{x},{y},{z}};

A1.X//MatrixForm

B = {{1},{-2},{3}};

E6.E5.E4.E3.E2.E1.A1

(Ainv=E6.E5.E4.E3.E2.E1)//MatrixForm

Ainv.A1//MatrixForm

B1 = {{11},{-2},{3}};

E6.E5.E4.E3.E2.E1.B1//MatrixForm

Ainv.B1//MatrixForm