For the system of linear equations
$\begin{array}{l} x + y + z = 6 \\ α x + β y + 7 z = 3 \\ x + 2 y + 3 z = 14 \end{array}$
Which of the following is NOT true?

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For every point $(α, β) \neq (7, 7)$ on the line $x - 2 y + 7 = 0$
the system has infinitely many solutions

If $α = β and α \neq 7$ then the system has a unique solution

If $α = β = 7$ , then system has no solution

There is a unique point $(α, β)$ on the line $x + 2 y + 18 = 0$ for
which the system has infinitely many solutions

answer is A.

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Detailed Solution

Consider
$|\begin{array}{ccc} 1 & 1 & 1 \\ α & β & 7 \\ 1 & 2 & 3 \end{array}| R_{3} \to R_{3} - R_{1} ~ |\begin{array}{ccc} 1 & 1 & 1 \\ α & β & 7 \\ 0 & - 1 & - 2 \end{array}|$
$\begin{array}{l} = (- 2 β + 7) - (- 2 α) + (- α) \\ = α - 2 β + 7 \end{array}$
If $(α, β)$ lies on $x - 2 y + 7 = 0$ then, we have determinant of coefficient matrix = 0
$(α, β) \neq (7, 7) \Rightarrow Δ \neq 0$ in many cases and we get a unique solution in those cases