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 [2023]

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Solution

Q.
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 [2023]

1 20

2 25

3 28

4 23

Ans.
(4)

Given system of equations are

$x + y + a z = b$

$2 x + 5 y + 2 z = 6$

$x + 2 y + 3 z = 3$

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

$Δ =$ $| \begin{matrix} 1 & 1 & a \\ 2 & 5 & 2 \\ 1 & 2 & 3 \end{matrix} |$ $= 0 \Rightarrow 1 (15 - 4) - 1 (6 - 2) + a (4 - 5) = 0$

$\Rightarrow 11 - 4 - a = 0 \Rightarrow a = 7$

$Δ_{1} =$ $| \begin{matrix} b & 1 & a \\ 6 & 5 & 2 \\ 3 & 2 & 3 \end{matrix} |$ $= 0$

$\Rightarrow b (15 - 4) - 1 (18 - 6) + a (12 - 15) = 0$

$\Rightarrow 11 b - 12 + 7 (- 3) = 0 \Rightarrow b = \frac{33}{11} = 3$

$∴$ $2 a + 3 b = 2 (7) + 3 (3) = 23$

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