Question

If the system of equations
$x-2y+3z=9$
$2x+y+z=b$
$x-7y+az=24,$ has infinitely many solutions, then $a-b$ is equal to

Answer 1

$D=0$
$\Rightarrow \left|\begin{array}{ccc}1& -2& 3\\ 2& 1& 1\\ 1& -7& a\end{array}\right|=0$
$\Rightarrow 1(a+7)+2(2a-1)+3(-14-1)=0$
$\Rightarrow a+7+4a-2-45=0$
$\Rightarrow 5a=40$
$\therefore a=8$
and $D_1=\left|\begin{array}{ccc}9& -2& 3\\ b& 1& 1\\ 24& -7& 8\end{array}\right|=0$
$\Rightarrow 9(8+7)+2(8b-24)+3(-7b-24)=0$
$\Rightarrow 135+16b-48-21b-72=0$
$\Rightarrow 15=5b\Rightarrow b=3$
$\therefore a-b=5$