The system of equations x+y+z=0, x-y+z=0, 2x+2z=0 has

A homogeneous system always has the trivial solution. Adding the first two equations gives $2x+2z=0$, which is the third equation, so the three equations are dependent and the determinant is zero. Subtracting gives $y=0$ and $x=-z$, a one-parameter family of solutions, hence infinitely many.