If \(\begin{bmatrix} x+y+z\\x+z \\ y + z\end{bmatrix} = \begin{bmatrix} 11\\6 \\ 8\end{bmatrix} \) then the value of x + 2y - 3z is

4

x + y + z = 11  ....(i)

x + z = 6  ....(ii)

y + z = 8 ....(iii)

Subtraction eq. (ii) form (iii)

y + z - (x + z) = 8 - 6

⇒ y + z - x - z = 2

⇒ y - x = 2

⇒ y = 2 + x

Putting y in eq. (i)

x + (2 + x) + z = 11

x + 2x + z = 11

2x + z = 11 - 2

2x + z = 9 ....(iv)

Subtraction eq. (iii) form (iv)

2x + z -(x + z) = 9 - 6

2x + z - x - z = 3

2x - x = 3

x = 3

y = 2 + x

y = 2 + 3 = 5

Put value of x in eq. (ii)

x + z = 6

3 + z = 6 ⇒ z = 6 - 3 = 3

 So, according to question x + 2y - 3z ⇒ 3 + 2 × (5) - 393)

= 3 + 10 - 9 = 3 + 1 = 4