System of equations (k+1)3x+(k+2)3y=(k+3)3 (k+1)x+(k+2)y=(k+3) x+y=1 is consistent. Since, the given system of equations are consistent. Then D=0 and Also, (D1=D2=D3=0) have infinitely many solutions. By Crammer Rule D=|
(k+1)3
(k+2)3
(k+3)3
(k+1)
(k+2)
(k+3)
1
1
1
|=0 ∵C2→C2−C1 C3→C3−C1 D=|
(k+1)3
(k+2)3−(k+1)3
(k+3)3−(k+1)3
(k+1)
(k+2)−(k+1)
(k+3)−(k+1)
1
0
0
| =0 D=|
(k+1)3
(3k2+9k+7)
(6k2+24k+26)
(k+1)
1
2
1
0
0
| =0 Expand with r to R3 2(3k2+9k+7)−2(3k2+12k+13)=0 ⇒3k2+9k+7−3k2−12k−13=0 ⇒−3k−6=0⇒3k=−6 ⇒k=−2