Question

Question 1 (8 marks) Consider the linear system x - 2y + 2z = -1 -2x + 3y + kz=1 2x + ky + (k - 4)2 = 1 (a) For which values of k is this system (i) consistent or (ii) inconsistent? (b) Find all solutions to the system when k = -1. (c) Describe your answer to (b) geometrically.

Answer 1

السا Date -24 ty tkz=1 extky Ce-WZEL Write Abore system of Equations to Ax= b where A-si-e a / - Ek ky] Det A= H 2 0 2 2k k-ul = 3 k 1 + 2l-ek 1+2 1-2 3 Kk-ul & k-ul = => 3 (4-4-2² +2 [-2K+ 8-247+2[-28-6)=0 = 3K-12-K²-8K+16 - 4K + /2=0 -k²-ak -8= 0 1 u2+96+ 8 = 0 - (K + 1) Che + 8 = 0 - 0° 0 K= 7 - 8 thus for k71-8 this yoten is Obriously consistent Hext we cher for k= 4, 8.

Date for k=-1 A = 1 - 2 ET 2 - - 5 Wnider the Acumented matrix & 1 2 T Ab]= I 3 - 1 R₂ - Ret2R, R3-) Rg 2R, 433 oo -9 3 I1 & 2-1 -1 +3 | 0 39 3 R3-) Rgt 3 k2 1 -2 2 -17 To 0 0 0 they ; (A; b) = f (A) = 2 => for k=1 systery is longstent

الالاDate Augmented snatox when k=- 8 then consider t-P 2 - 2 3 - 8 2 -8 -12 -17 1 1 R2 R2 +2R1 R3 R3-2R, u 1-2 247 O + -4 - Lo-4-163 - Ry -> R3-4R2 ur -2 2-17 Lo o o z thus f (4 b) = 37 P (A) = 2 do system i Inwontent for kare systim is consistent for K7-8 system is Inconsistent for k=-8

الالاDate for k=-1 We have L x- sy +22=7 - 0 23-49-521 - Now solving a ly -3x+4y=- x = 4y - 1 - 1 y 1 Ad dorg (y & (3) ey - 82=2 y = 1432 x = (1+32) - / x = 42-| 2 in pul - 3-Tuple (42-1, (1 + 32) 2.2tll form solution alt g 4x = b system Geometrically represent a line paoning through A 1, 0). = (o