PrecalculusSuperchallengeSystems, Matrices, and Determinants

Systems, Matrices, and Determinants · Superchallenge problem

For how many ordered triples $(a,b,c)$ of integers with $1\le a,b,c\le30$ does the system

x+y+2z=a,\qquad 2x-y+z=b,\qquad 4x+y+5z=c

have at least one solution $(x,y,z)$ in real numbers?