If system of equation ax + y + z = a, x + by + z = b and x + y + cz...

We have,&

$ax + y + z = a$~ & $. . . ( i )$

$x + by + z = b$~ & $. . . ( ii )$

$x + y + cz = c$~ & $. . . ( iii )$

are inconsistent.~

\begin{cases}∴ & a 1 1 \\ 1 & b & 1 \\ 1 & 1 & c\end{cases} = 0and\left|\begin{cases}a & 1 & 1 \\ b & b & 1 \\ c & 1 & c\end{cases}\right| \ne 0

⇒ \left|\begin{cases}a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c\end{cases}\right| = 0

$⇒ a ( bc - 1 ) - 1 ( c - 1 ) + 1 ( 1 - b ) = 0$

$abc - a - c + 1 + 1 - b = 0$

$abc - a - b - c + 2 = 0$

and~\left|$\begin{cases}a & 1 & 1 \\ b & b & 1 \\ c & 1 & c\end{cases}$\right| \ne 0$</p><p>It is possible only when$a \ne 1, b \ne 1$and$c \ne 1