First slide

System of linear equations

Question

$The values of λ and μ for which the system of linear equations$
$\begin{array}{l} x + y + z = 2 \\ x + 2 y + 3 z = 5 \\ x + 3 y + λ z = μ \end{array}$
has infinitely many solutions are, respectively :

Moderate

A

4 and 9
B

5 and 7
C

6 and 8
D

5 and 8

Solution

$\begin{array}{l} \begin{matrix} x + y + z = 2 & (1) \\ x + 2 y + 3 z = 5 & (2) \end{matrix}\} has infinitely - many solutions \\ x + 3 y + λ z = m \\ (2) - (1) gives y + 2 z = 3 \to (4) hasinfinitely - many solutions \\ (3) - (2) gives y + (λ - 3) z = μ - 5 \to (5)} \\ \Rightarrow \frac{1}{1} = \frac{λ - 3}{2} = \frac{μ - 5}{3} \\ \Rightarrow λ = 5 and μ = 8 \end{array}$

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