Let x,y∈R, such that the set of equations x+(1+k)y=0(1−k)x+ky=k+1(1+k)x+(12−k)y=− k−1is consistent Then, which of the following is/are CORRECT statement(s)? (k∈R)

Adding 2nd  and 3rd  equations: 2x+12y=0 ⇒x+6y=0 Also, x+(1+k)y=0⇒ If y=0,x=0⇒k=−1 Or k+1=6⇒k=5k=5 gives: x+6y=0−4x+5y=66x+7y=−6 Clearly consistent with solution being x=−3629y=62910x+2y=−125x+y=−6−30y+y=6y=629