If the system of linear equations x - 2 y + z = - 4 2 x + α y + 3 z = 5 3 x - y + β z = 3 has infinitely many solutions, then 12 α + 13 β is equal to

Step-by-step solution

Given: The system of linear equations x - 2 y + z = - 4 , 2 x + α y + 3 z = 5 and 3 x - y + β z = 3 has infinity many solutions, So, △ = △ 1 = △ 2 = △ 3 Now, finding, △ 2 = 1 - 4 1 2 5 3 3 3 β = 0 ⇒ 5 β - 9 + 8 β - 36 - 9 = 0 ⇒ 13 β = 54 And, △ 3 = 1 - 2 - 4 2 α 5 3 - 1 3 = 0 ⇒ 3 α + 5 + 12 - 30 + 8 + 12 α = 0 ⇒ 15 α = 5 ⇒ 12 α = 4 Hence, the value of 12 α + 13 β = 4 + 54 = 58