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:
$D=\left|\begin{array}{ccc}1 & -2 & 3 \\ 2 & 1 & 1 \\ 1 & -7 & a\end{array}\right|=0 \Rightarrow a=8$
also, $D_{1}=\left|\begin{array}{ccc}9 & -2 & 3 \\ b & 1 & 1 \\ 24 & -7 & 8\end{array}\right|=0 \Rightarrow b=3$
hence, $a-b=8-3=5$