Question:
If the system of equations $x-2 y+3 z=9,2 x+y+z=b$
$x-7 y+a z=24$, has infinitely many solutions, then $a-b$ is equal to__________.
Solution:
For infinitely many solutions,
$\Delta=\Delta_{1}=\Delta_{2}=\Delta_{3}=0$
$\Delta=\left|\begin{array}{ccc}1 & -2 & 3 \\ 2 & 1 & 1 \\ 1 & -7 & a\end{array}\right|=0$
$\Rightarrow(a+7)-2(1-2 a)+3(-15)=0$
$\Rightarrow a=8$
$\Delta_{3}=\left|\begin{array}{ccc}1 & -2 & 9 \\ 2 & 1 & b \\ 1 & -7 & 24\end{array}\right|=0$
$\Rightarrow(24+7 b)-2(b-48)+9(-15)=0$
$\Rightarrow b=3$
$\therefore a-b=5$