z=4, y=-25, x=18 

These do not check in the first equation

  3x  +   y   +  z = 0
3(18) + (-25) +  4 = 0
  54  -   25  +  4 = 0
                33 = 0

These do not check in the second equation, either.

  2x  -    y  +  z = 0
2(18) - (-25) +  4 = 0
  36  +   25  +  4 = 0
                65 = 0

These do not check in the third equation, either.

  2x  +    y  +  z = 0
2(18) + (-25) +  4 = 0
  36  -   25  +  4 = 0
                15 = 0

The only solution is x=0, y=0, z=0

The other tutor is right in one respect, but wrong in another respect.

When the numbers on the right (not coefficients, as he stated) are all
aero, it's true that x = y = z = 0 is a solution.  However that is not
necessarily the only solution.  It was in your case, but not in all
cases.

For example:

x +  y + 3z = 0
x + 2y + 4z = 0
x + 3y + 5z = 0

This one has infinitely many solutions.

x = y = z = 0  is indeed a solution, but also

x=-2, y=-1, and z=1 is a solution and

x=4, y=2, and z=-2 is also a solution.

The general solution is

x = -2k, y = -k, and z = k, where k is any number

Edwin