First slide

System of linear equations

Question

If the system of equations $x + y + z = 5, x + 2 y + 3 z = 9$ , $x + 3 y + α z = β$ has infinitely many solutions , then $β - α$ equals

Moderate

A

8
B

18
C

21
D

5

Solution

Since, the system of equations has infinitely many solution, therefore
$D = D_{1} - D_{2} - D_{3} = 0$
Here,
$\begin{array}{l} D = |\begin{array}{lll} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & α \end{array}| = 1 (2 α - 9) - 1 (α - 3) + 1 (3 - 2) \\ = α - 5 \end{array}$
$And D_{3} = |\begin{array}{lll} 1 1 5 \\ 1 2 9 \\ 1 3 β \end{array}| = 1 (2 β - 27) - 1 (β - 9) + 5 (3 - 2)$
and $D_{3} = |\begin{matrix} 1 & 1 & 5 \\ 1 & 2 & 9 \\ 1 & 3 & β \end{matrix}| = 1 (2 β - 27) - 1 (β - 9) + 5 (3 - 2)$
$= β - 13$
$Now, D = 0$
$\Rightarrow α - 5 = 0 \Rightarrow α = 5$
$And D_{3} = 0 \Rightarrow β - 13 = 0$
$\begin{array}{l} \Rightarrow β = 13 \\ ∴ β - α = 13 - 5 = 8 \end{array}$

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