If the system of linear equations 7 x + 11 y + z = 13 , 5 x + 4 y + 7 z = , 175 x + 194 y + 57 z = 361 has infinitely many solutions, then 2 is equal to: [2023]

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Solution

Q.
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: [2023]

1 4

2 3

3 6

4 5

Ans.
(1)

For infinitely many solutions, $Δ = Δ_{1} = Δ_{2} = Δ_{3} = 0$

$Δ =$ $| \begin{matrix} 7 & 11 & α \\ 5 & 4 & 7 \\ 175 & 194 & 57 \end{matrix} |$

$= 7 (228 - 1358) - 11 (285 - 1225) + α (970 - 700)$

$= - 7910 + 10340 + 270 α = 0 \Rightarrow - 2430 + 270 α = 0 \Rightarrow α = - 9$

$Δ_{1} =$ $| \begin{matrix} 13 & 11 & - 9 \\ β & 4 & 7 \\ 361 & 194 & 57 \end{matrix} |$ $= 13 (228 - 1358) - 11 (57 β - 2527) - 9 (194 β - 1444)$

$= - 14690 - 627 β + 27797 - 1746 β + 12996 = 26103 - 2373 β = 0$

$\Rightarrow β = 11$ $∴ α + β + 2 = - 9 + 11 + 2 = 4$

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