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:

$D=\left|\begin{array}{ccc}1 & -2 & 3 \\ 2 & 1 & 1 \\ 1 & -7 & a\end{array}\right|=0 \Rightarrow a=8$

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

hence, $a-b=8-3=5$

 

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