Consider the following system of equations:x+2y-3z=a2x+6y-11z=bx-2y+7z=cWhere a,b and c are real constants. Then the system of equations:

x+2y-3z=a 2x+6y-11z=b x-2y+7z=cΔ=12-326-111-27=0 Δ1=a2-3b6-11c-27=45a-2b-cΔ2=1a-32b-111c7=-55a-2b-c Δ3=12a26b1-2c=-25a-2b-c . If 5a=2b+c⇒Δ1=Δ2=Δ3=0⇒ Infinitely many solutions

Consider the following system of equations:x+2y-3z=a2x+6y-11z=bx-2y+7z=cWhere a,b and c are real constants. Then the system of equations:

x+2y-3z=a 2x+6y-11z=b x-2y+7z=cΔ=12-326-111-27=0 Δ1=a2-3b6-11c-27=45a-2b-cΔ2=1a-32b-111c7=-55a-2b-c Δ3=12a26b1-2c=-25a-2b-c . If 5a=2b+c⇒Δ1=Δ2=Δ3=0⇒ Infinitely many solutions

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Consider the following system of equations:
$\begin{array}{l} x + 2 y - 3 z = a \\ 2 x + 6 y - 11 z = b \\ x - 2 y + 7 z = c \end{array}$
Where a,b and c are real constants. Then the system of equations:

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$has infinite number of solutions when 5 a = 2 b + c$

has no solution for all a,b and c

has a unique solution for all a,b and c

$has a unique solution when 5 a = 2 b + c$

answer is C.

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Detailed Solution

$\begin{array}{l} x + 2 y - 3 z = a 2 x + 6 y - 11 z = b x - 2 y + 7 z = c \\ Δ = |\begin{matrix} 1 & 2 & - 3 \\ 2 & 6 & - 11 \\ 1 & - 2 & 7 \end{matrix}| = 0 Δ_{1} = |\begin{matrix} a & 2 & - 3 \\ b & 6 & - 11 \\ c & - 2 & 7 \end{matrix}| = 4 (5 a - 2 b - c) \\ Δ_{2} = |\begin{matrix} 1 & a & - 3 \\ 2 & b & - 11 \\ 1 & c & 7 \end{matrix}| = - 5 (5 a - 2 b - c) Δ_{3} = |\begin{matrix} 1 & 2 & a \\ 2 & 6 & b \\ 1 & - 2 & c \end{matrix}| = - 2 (5 a - 2 b - c) \\ . If 5 a = 2 b + c \Rightarrow Δ_{1} = Δ_{2} = Δ_{3} = 0 \Rightarrow Infinitely many solutions \end{array}$