The number of solutions for the system of equations $x+y-z=0,3x-y-z=0,x-3y+z=0$ is

The given system of equations is 

$x+y-z=0,3x-y-z=0,x-3y+z=0$

This is homogenous system of equations. It has either trivial solutions or non trivial  solutions

If the determinant of the coefficient matrix is zero then the system of homogeneous equations has nontrivial solutions, if the determinant of the coefficient matrix is nonzero, then the system of homogeneous equations has trivial solutions. 

Consider the determinant of the coefficient matrix: 

For the system of given homogeneous equations 

$\begin{array}{rcl}\mathrm{\Delta }& =& \left|\begin{array}{rrr}1& 1& -1\\ 3& -1& -1\\ 1& -3& 1\end{array}\right|\\ & =& 1(-1-3)-1(3+1)-1(-9+1)\\ & =& -4-4+8\\ & =& 0\end{array}$

Since the determinant of the coefficient matrix is zero, the system of homogeneous equations has nontrivial solutions, it is infinite number of solutions.