If the following system of linear equations 2x + y + z = 5 x - y +...

Given system of equations

D = \left|\begin{cases}2 & 1 & 1 \\ 1 & - 1 & 1 \\ 1 & 1 & a\end{cases}\right| = 0 ⇒ - 2a - 2 - a + 1 + 2 = 0 ⇒ a = \dfrac{1}{3}

{D}_{1} = \left|\begin{cases}5 & 1 & 1 \\ 3 & - 1 & 1 \\ b & 1 & a\end{cases}\right| = 5\left(- a - 1\right) + - 1\left(3a - b\right) + 1\left(3 + b\right) = - 8a + 2b - 2

{D}_{2} = \left|\begin{cases}2 & 5 & 1 \\ 1 & 3 & 1 \\ 1 & b & a\end{cases}\right| = 2\left(3a - b\right) - 5\left(a - 1\right) + 1\left(b - 3\right) = a - b + 3

{D}_{3} = \left|\begin{cases}2 & 1 & 5 \\ 1 & - 1 & 3 \\ 1 & 1 & b\end{cases}\right| = 2\left(- b - 3\right) - 1\left(b - 3\right) + 5\left(2\right) = 7 - 3b

For the system to have no solution $⇒ D = 0, {D}_{3} \ne 0$

$∴ a = \dfrac{1}{3}, b \ne \dfrac{7}{3}$.