We create a matrix A, and then row-reduce it.

A={{2,4,-9,10,9,-14,15}, {4,8,-18,20,22,-34,38},{2,4,-3,4,7,18,-13},
{10,20,-30,35,50,2,22},{-2,-4,6,-7,-6,-5,3}}

A//MatrixForm

R=RowReduce[A] // MatrixForm

Now we row-reduce manually.

A1=A

A1[[1]]=(1/2)*A1[[1]] ; A1//MatrixForm

A2=A1; A2[[2]]=A2[[2]]-4*A2[[1]]; A2//MatrixForm

A3=A2; A3[[3]]=A3[[3]]-2*A3[[1]]; A3//MatrixForm

A4=A3; A4[[4]]=A4[[4]]-10*A4[[1]]; A4//MatrixForm

A5=A4; A5[[5]]=A5[[5]]+2*A5[[1]]; A5//MatrixForm

A6=A5; A6[[{2,3}]]=A6[[{3,2}]]; A6//MatrixForm

A7=A6; A7[[2]]=(1/6)*A7[[2]]; A7//MatrixForm

A8=A7; A8[[1]]=A8[[1]]+(9/2)*A8[[2]]; A8[[4]]=A8[[4]]-15*A8[[2]];A8[[5]]=A8[[5]]+3*A8[[2]]; A8//MatrixForm

A9=A8;A9[[3]]=(1/4)*A9[[3]]; A9//MatrixForm

A10=A9;A10[[1]]=A10[[1]]-3*A10[[3]];A10[[2]]=A10[[2]]+(1/3)*A10[[3]];A10[[4]]=A10[[4]]-10*A10[[3]];A10[[5]]=A10[[5]]-2*A10[[3]];A10//MatrixForm

A11=A10; A11[[4]]=(1/7)*A11[[4]]; A11//MatrixForm

A12=A11; A12[[1]]=A12[[1]]-A12[[1,6]]*A12[[4]]; A12[[2]]=A12[[2]]-A12[[2,6]]*A12[[4]];A12[[3]]=A12[[3]]-A12[[3,6]]*A12[[4]];A12//MatrixForm

R

We generate a matrix from some equations, and then solve them.

e1=3*x-11*y+7*z==2; e2=x-4*y+8*z==3; e3=2*x-5*y-4*z==11;

Eq={e1,e2,e3}

Solve[Eq,{x,y,z}]

s=Normal[CoefficientArrays[Eq,{x,y,z}]]

cm=s[[2]]

cm//MatrixForm

We form the augmented matrix.

s[[1]]

am=Transpose[Append[Transpose[cm],s[[1]]]]

am //MatrixForm

RowReduce[am] //MatrixForm