If the system of equations
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,

$\Delta=\Delta_{1}=\Delta_{2}=\Delta_{3}=0$

$\Delta=\left|\begin{array}{ccc}1 & -2 & 3 \\ 2 & 1 & 1 \\ 1 & -7 & a\end{array}\right|=0$

$\Rightarrow(a+7)-2(1-2 a)+3(-15)=0$

$\Rightarrow a=8$

$\Delta_{3}=\left|\begin{array}{ccc}1 & -2 & 9 \\ 2 & 1 & b \\ 1 & -7 & 24\end{array}\right|=0$

$\Rightarrow(24+7 b)-2(b-48)+9(-15)=0$

$\Rightarrow b=3$

$\therefore a-b=5$