If the system of equations

$x-2y + 3z = 9 $

$2x+ y + z= b $

$x-ty + az = 24 $

has infinitely many solutions, then $a-b $ is equal to

5

The correct answer is option (2) : 5

If the given system of equations has infinitely many solutions, then

$D= D_1=D_2=D_3=0$

Now, $

$D=0 ⇒\begin{vmatrix} 1 & -2 & 3 \\ 2 & 1 & 1\\ 1 & -7 & a \end{vmatrix}= 0 $

$⇒(a+7) +2(2a-1)+3(-14-1)=0$

$⇒a+7 +4a -2 -45 =0 ⇒ 5a -40 =0 ⇒ a = 8 $

$D_1=0 ⇒ \begin{vmatrix} 9 & -2 & 3 \\ b & 1 & 1\\ 24 & -7 & a \end{vmatrix}=0⇒\begin{vmatrix} 9 & -2 & 3 \\ b & 1 & 1\\ 24 & -7 & 8 \end{vmatrix}=0$

$⇒135+2(8b-24)+3(-7b-24)=0$

$⇒135+16b-48 -21b -72 =0 $

$⇒-5b +15 =0 ⇒b =3.$

For these values of a and b, we find that $D_2=D_3=0 $ and $a-b = 8-3= 5 $