If the system of linear equations 7 x + 11 y + α z = 13 5 x + 4 y + 7 z = β 175 x + 194 y + 57 z = 361 has infinitely many solutions, then α + β + 2 is equal to

Step-by-step solution

Given, System of linear equations, 7 x + 11 y + α z = 13 5 x + 4 y + 7 z = β 175 x + 194 y + 57 z = 361 Now we know that, for infinite solution D = D 1 = D 2 = D 3 = 0 Now solving D = 0 we get, 7 11 α 5 4 7 175 194 57 = 0 Now applying operation R 3 → R 3 - 25 R 1 we get, 7 11 α 5 4 7 0 - 81 57 - 25 α = 0 ⇒ 81 ( 49 - 5 α ) + ( 57 - 25 α ) ( - 27 ) = 0 ⇒ 270 α = - 2430 ⇒ α = - 9 And now solving D 1 = 0 we get, 13 11 - 9 β 4 7 361 194 57 = 0 &#8