The system of linear equations x−y+z=1, x+y−z=3, x−4y+4z=αhas

givenx−y+z=1______1x+y−z=3______2x−4y+4z=α______3From eqns 1 & 2     x=2⇒y-z=1 & −4y+4z=α−24y−4z+−4y+4z=4+α−2⇒α+2=0⇒α=−2The system of has no solution forα≠−2   If α=−2, then the system is consistent and has infinite number of solutions. (since y-z=1 has infinite number of solutions)