If the system of equations 2 x + 3 y − z = 5 x + α y + 3 z = − 4 3 x − y + β z = 7 has infinitely many solutions, then 13 α β is equal to

Step-by-step solution

Given: The system of equations, 2 x + 3 y - z = 5 x + α y + 3 z = - 4 3 x - y + β z = 7 have many solutions, So, △ = △ 1 = △ 2 = △ 3 Now, finding △ 2 = 2 5 - 1 1 - 4 3 3 7 β = 0 ⇒ 2 - 4 β - 21 - 5 β - 9 - 1 7 + 12 = 0 ⇒ 13 β = - 16 Now, finding △ 3 = 2 3 5 1 α - 4 3 - 1 7 = 0 ⇒ 14 α - 8 - 57 - 5 - 15 α = 0 ⇒ α = - 70 Hence, the value of 13 α β = 70 × 16 = 1120