Do you know anything about matrices? If so you can view this system of equations (called homogenous because we set them all equal to 0) as a matrix.

c1 c2 c3

c4 c5 c6

c7 c8 c9

It is a fact that these equations have nontrivial solution when set equal to 0 (the trivial solution is if you set x=y=z=0) if and only if the determinant of the matrix is 0. If the determinant is 0 there is a way to come up with a general formula for solutions. Look up Gauss-Jordan elimination.