How do I determine whether a system of 3 linear equations is consistent or not?

First, form the system into a 3x3 matrix using the coefficients. Find the determinant of this matrix.If the determinant =/= 0, then the matrix is singular and has a unique solution. The system is consistent and the planes coincide at a point.
If the determinant is 0, then the matrix is non-singular, so use simultaneous equations to attempt to solve the system:
If the system contains 3 equations that are multiples of each other, then the system is consistent and represent a single plane.
If the resolved system gives a redundant equation after elimination/substitution, then the system is consistent and represents a sheaf. The planes coincide at a line.
In all cases where the system is consistent, the system can be resolved to give a set of solutions, whether that be a single point, a line or a plane.