Consider the following system of equations: x + 2y - 3z = a 2x + 6y...

D = \left|\begin{cases}1 & 2 & - 3 \\ 2 & 6 & - 11 \\ 1 & - 2 & 7\end{cases}\right|

$= 20 - 2 ( 25 ) - 3 ( - 10 )$

$= 20 - 50 + 30 = 0$

{D}_{1} = \left|\begin{cases}a & 2 & - 3 \\ b & 6 & - 11 \\ c & - 2 & 7\end{cases}\right|

$= 20a - 2 ( 7b + 11c ) - 3 ( - 2b - 6c )$

$= 20a - 14b - 22c + 6b + 18c$

$= 20a - 8b - 4c$

$= 4 ( 5a - 2b - c )$

{D}_{2} = \left|\begin{cases}1 & a & - 3 \\ 2 & b & - 11 \\ 1 & c & 7\end{cases}\right|

$= 7b + 11c - a ( 25 ) - 3 ( 2c - b )$

$= 7b + 11c - 25a - 6c + 3b$

$= - 25a + 10b + 5c$

$= - 5 ( 5a - 2b - c )$

{D}_{3} = \left|\begin{cases}1 & 2 & a \\ 2 & 6 & b \\ 1 & - 2 & c\end{cases}\right|

$= 6c + 2b - 2 ( 2c - b ) - 10a$

$= - 10a + 4b + 2c$

$= - 2 ( 5a - 2b - c )$

for infinite solution

$D = {D}_{1} = {D}_{2} = {D}_{3} = 0$

$⇒ 5a = 2b + c$