4x-3y+z=0, -2x-y+3z=0, -14+3y-7z=0

I Want to find : Rank (linear algebra) with Gaussian Method!

Solving these in my head gives x = -4, y = -7, z = -5, so the rank must be 3. (if we can solve the equations fully like this, they're independent)
If I remember correctly, the Gaussian method means reducing it to diagonal form with elementary row operations. So first add 14 to both sides of equation 3, then write the equations as rows of coefficients:

4 .. -3 .. 1 .. 0
-2 .. -1 .. 3 .. 0
0 .. 3 . -7 . 14

Add 2*row2 to row1:

0 .. -5 .. 7 .. 0
-2 .. -1 .. 3 .. 0
0 .. 3 . -7 . 14

Add row3 to row1:
0 .. -2 .. 0 .. 14
-2 .. -1 .. 3 .. 0
0 .. 3 . -7 . 14

Multiply row3 by 2, then add 3*row1 to row3:
0 .. -2 .. 0 .. 14
-2 .. -1 .. 3 .. 0
0 .. 0 . -14 . 70

Divide row 1 by -2, swap positions of rows 1 and 2, divide row3 by -14:
-2 .. -1 .. 3 .. 0
0 .. 1 ... 0 .. -7
0 .. 0 .. 1 ... -5

Add -3*row3 + row2 to row1:
-2 .. 0 .. 0 .. 8
0 .. 1 .. 0 .. -7
0 .. 0 .. 1 .. -5