If the system of equations x + y + a z = b 2 x + 5 y + 2 z = 6 x + 2 y + 3 z = 3 has infinitely many solutions, then 2 a + 3 b is equal to

Step-by-step solution

Given system of equations are x + y + a z = b . . . . . i 2 x + 5 y + 2 z = 6 . . . . . ii x + 2 y + 3 z = 3 . . . . . iii It is given that the system of equations have infinite solutions. ⇒ ∆ = 0 ⇒ 1 1 a 2 5 2 1 2 3 = 0 ⇒ 1 15 - 4 - 1 6 - 2 + a 4 - 5 = 0 ⇒ a = 7 Also, we have ⇒ ∆ 1 = 0 ⇒ b 1 a 6 5 2 3 2 3 = 0 ⇒ b 15 - 4 - 1 18 - 6 + 7 12 - 15 = 0 ⇒ b = 3 ∴ a = 7 ,   b = 3 ∴ 2 a + 3 b = 14 + 9 = 23 Hence, this is the correct option.