The system of linear equations x + y + z = 6 2 x + 5 y + a z = 36 x + 2 y + 3 z = b has [2026]

Question

The system of linear equations

$\begin{matrix} x + y + z = 6 \\ 2 x + 5 y + a z = 36 \\ x + 2 y + 3 z = b \end{matrix}$ has [2026]

Smart Achievers · Accepted Answer

(3)

$If D =$ $| \begin{matrix} 1 & 1 & 1 \\ 2 & 5 & a \\ 1 & 2 & 3 \end{matrix} |$ $= 0 \Rightarrow a = 8$

$If D_{1} =$ $| \begin{matrix} 6 & 1 & 1 \\ 36 & 5 & a \\ b & 2 & 3 \end{matrix} |$ $= 0 \Rightarrow a b - 5 b - 12 a + 54 = 0$

$If D_{2} =$ $| \begin{matrix} 1 & 6 & 1 \\ 2 & 36 & a \\ 1 & b & 3 \end{matrix} |$ $= 0 \Rightarrow a b - 6 a - 2 b - 36 = 0$

$If D_{3} =$ $| \begin{matrix} 1 & 1 & 6 \\ 2 & 5 & 36 \\ 1 & 2 & b \end{matrix} |$ $= 0 \Rightarrow b = 14$

$For a = 8 and b = 14 \Rightarrow D_{1}, D_{2} are also zero$

$For a = 8 and b = 14 \Rightarrow D = D_{1} = D_{2} = D_{3} = 0$

$\Rightarrow infinitely many solutions.$

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